Wikiversity:Colloquium

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

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

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

== Proposal to rehost Wikinews here ==

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

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

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

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

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

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

== Create an autopatrolled user group? ==

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

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

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

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

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

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

== Draft inactivity policy ==

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

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

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)

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

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

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

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

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

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

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

== Towards an Ethics policy ==

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

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

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

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

Hello community,

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

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

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

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

== Image not displaying ==

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

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[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; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &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; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}</text>
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      <comment>Reverted edits by [[Special:Contributions/Farhan Kamal Azeemi 133|Farhan Kamal Azeemi 133]] ([[User_talk:Farhan Kamal Azeemi 133|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <minor />
      <comment>Reverted edit by [[Special:Contributions/~2026-35901-15|~2026-35901-15]] ([[User_talk:~2026-35901-15|talk]]) to last version by [[User:Atcovi|Atcovi]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="50" sha1="phlij3i0lq7l17sacctmpzowd8epftu" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}</text>
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      <parentid>2812881</parentid>
      <timestamp>2026-06-20T16:32:31Z</timestamp>
      <contributor>
        <username>~2026-35959-56</username>
        <id>3095905</id>
      </contributor>
      <comment>Added more accurate information.</comment>
      <origin>2816352</origin>
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      <text bytes="1381" sha1="4deby2652yfz1oojx1h6lfjnbjp3606" xml:space="preserve">There are a great variety of other instruments that have been made that are no longer around.
Here are some examples:

*Basset Horn: A single reed instrument that was often bent in a variety of shapes then sent through a soundbox
*Natural Horn: An earlier form of the [[wikipedia:French_horn|French Horn]] which uses a series of tubing extensions which attach at the mouthpiece hole.
*Rackett: A double-reed instrument that sends the sound through a series of waves to deepen the sound.
*Saxhorn: Adolphe Sax developed both the woodwind saxophones, and an odd variety of brass instruments called the saxhorn, a member of the [[wikipedia:Aerophone|Aerophone]] family of instruments.
*Theremin: An electronic instrument that features two antennas, and is played by moving one's hands up and down and side-to-side.
*Valve Trombone: A trombone that uses valves, like a trumpet, to vary the pitch, but still makes the sound of a trombone.
*Slide Trumpet: Essentially the opposite of the aforementioned valve trombone.
*Viola da Gamba: An early form of the [[wikipedia:Violin|Violin]], which features 6 or 7 strings instead of 4.
*Wagner Tuba: A four-valve brass instrument created by [[w:Richard Wagner|Richard Wagner]], similar to the tuba, but built more in a manner similar to the French Horn.

There are a great many others, but this is just a start.

[[Category:Music instruments]]</text>
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  </page>
  <page>
    <title>Meher Baba</title>
    <ns>0</ns>
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      <parentid>1422976</parentid>
      <timestamp>2026-06-20T12:05:52Z</timestamp>
      <contributor>
        <username>~2026-35928-06</username>
        <id>3095852</id>
      </contributor>
      <comment>Updated links to current locations</comment>
      <origin>2816326</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2790" sha1="55e9n2j22f95w17faup1d8xvinzuw9r" xml:space="preserve">[[Image:Meher_Baba_1957.jpg|thumb|Meher Baba]]
Welcome to the '''Center for the Study of Meher Baba''', part of the [[School:Theology|School of Theology]] and the [[Topic:Religious Studies|Division of Religious Studies]].

==Center Description==
The Wikiversity Center for the Study of [[w:Meher Baba|Meher Baba]] is a content development project where participants create, organize and develop Wikiversity content about Meher Baba.

== Learning Resources ==
* [[/Teachings and methodology/]]: an overview of Meher Baba's teachings and methods.

==External Learning Resources==

Fortunately there is a large body of source material by and about Meher Baba freely available on the internet. Many of Baba's major books can be searched on line or downloaded free in PDF format for research purposes. You are encouraged to make yourself familiar with what is freely available. Note that these materials are copyrighted by the Avatar Meher Baba Perpetual Public Charitable Trust. They are exclusively for research and educational purposes, and not for republication or resale. 

* [https://avatarmeherbabatrust.org/online-library-2/ The Trust Online Library] 
** ''The Trust Online Library'' has books by and about Meher Baba available to download as [[w:Portable Document Format|PDF]] files 
* [https://www.lordmeher.org/rev/index.jsp Lord Meher] 
** This 6,742 page fully searchable biography of Meher Baba was written by Bhau Kalchuri shortly after Baba's death in 1969 and was published in a 20 volume set in 1986. It is based on journals kept by Meher Baba's followers from as early as 1922, as well as hundreds of hours of recorded interviews. 
* [http://discoursesbymeherbaba.org Discourses by Meher Baba] 
** The 1967 6th edition of Meher Baba's ''Discourses''. While some of these discourses were dictated verbatim by Meher Baba, others were worked up from points given by Baba to his close disciple Dr. C. D. Deshmukh, an Indian professor of philosophy. The discourses originally appeared in ''Meher Baba Journal'' from 1938 to 1943 and were later gathered as books, receiving numerous editions and revisions.
* [http://www.theawakenermagazine.org/ Awakener Magazine Archive] 
** ''Awakener Magazine'' was a magazine exclusively on Meher Baba that was published by Filis Frederick of California from 1953-1986. All 67 issues are now available online and are searchable. 
* [http://www.ambppct.org/messages.php Messages of Meher Baba]
** Messages given out by Meher Baba over the course of his life, often in the form of circulars or pamphlets.
* [http://www.avatarmeherbaba.org/erics/glossary.html Glossary of Meher Baba's terminology]
** An essential resource for studying the works of Meher Baba, this glossary defines terms as Meher Baba used them.


[[Category:Meher Baba Studies]]</text>
      <sha1>55e9n2j22f95w17faup1d8xvinzuw9r</sha1>
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    <revision>
      <id>2816327</id>
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        <id>243757</id>
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      <text bytes="2782" sha1="el2r1abqmyc6po93266l4vugw98sz17" xml:space="preserve">[[Image:Meher_Baba_1957.jpg|thumb|Meher Baba]]
Welcome to the '''Center for the Study of Meher Baba''', part of the [[School:Theology|School of Theology]] and the [[Topic:Religious Studies|Division of Religious Studies]].

==Center Description==
The Wikiversity Center for the Study of [[w:Meher Baba|Meher Baba]] is a content development project where participants create, organize and develop Wikiversity content about Meher Baba.

== Learning Resources ==
* [[/Teachings and methodology/]]: an overview of Meher Baba's teachings and methods.

==External Learning Resources==

Fortunately there is a large body of source material by and about Meher Baba freely available on the internet. Many of Baba's major books can be searched on line or downloaded free in PDF format for research purposes. You are encouraged to make yourself familiar with what is freely available. Note that these materials are copyrighted by the Avatar Meher Baba Perpetual Public Charitable Trust. They are exclusively for research and educational purposes, and not for republication or resale. 

* [https://avatarmeherbabatrust.org/online-library-2/ The Trust Online Library] 
** ''The Trust Online Library'' has books by and about Meher Baba available to download as [[w:Portable Document Format|PDF]] files 
* [https://www.lordmeher.org/rev/index.jsp Lord Meher] 
** This 6,742 page fully searchable biography of Meher Baba was written by Bhau Kalchuri shortly after Baba's death in 1969 and was published in a 20 volume set in 1986. It is based on journals kept by Meher Baba's followers from as early as 1922, as well as hundreds of hours of recorded interviews. 
* [http://discoursesbymeherbaba.org Discourses by Meher Baba] 
** The 1967 6th edition of Meher Baba's ''Discourses''. While some of these discourses were dictated verbatim by Meher Baba, others were worked up from points given by Baba to his close disciple Dr. C. D. Deshmukh, an Indian professor of philosophy. The discourses originally appeared in ''Meher Baba Journal'' from 1938 to 1943 and were later gathered as books, receiving numerous editions and revisions.
* [http://www.theawakenermagazine.org/ Awakener Magazine Archive] 
** ''Awakener Magazine'' was a magazine exclusively on Meher Baba that was published by Filis Frederick of California from 1953-1986. All 67 issues are now available online and are searchable. 
* [http://www.ambppct.org/messages.php Messages of Meher Baba]
** Messages given out by Meher Baba over the course of his life, often in the form of circulars or pamphlets.
* [https://avatarmeherbabatrust.org/messages/ Glossary of Meher Baba's terminology]
** An essential resource for studying the works of Meher Baba, this glossary defines terms as Meher Baba used them.


[[Category:Meher Baba Studies]]</text>
      <sha1>el2r1abqmyc6po93266l4vugw98sz17</sha1>
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    <revision>
      <id>2816328</id>
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      <text bytes="2774" sha1="6oxcc9c3qrjo80rr0w8yp5tir0kdkrw" xml:space="preserve">[[Image:Meher_Baba_1957.jpg|thumb|Meher Baba]]
Welcome to the '''Center for the Study of Meher Baba''', part of the [[School:Theology|School of Theology]] and the [[Topic:Religious Studies|Division of Religious Studies]].

==Center Description==
The Wikiversity Center for the Study of [[w:Meher Baba|Meher Baba]] is a content development project where participants create, organize and develop Wikiversity content about Meher Baba.

== Learning Resources ==
* [[/Teachings and methodology/]]: an overview of Meher Baba's teachings and methods.

==External Learning Resources==

Fortunately there is a large body of source material by and about Meher Baba freely available on the internet. Many of Baba's major books can be searched on line or downloaded free in PDF format for research purposes. You are encouraged to make yourself familiar with what is freely available. Note that these materials are copyrighted by the Avatar Meher Baba Perpetual Public Charitable Trust. They are exclusively for research and educational purposes, and not for republication or resale. 

* [https://avatarmeherbabatrust.org/online-library-2/ The Trust Online Library] 
** ''The Trust Online Library'' has books by and about Meher Baba available to download as [[w:Portable Document Format|PDF]] files 
* [https://www.lordmeher.org/rev/index.jsp Lord Meher] 
** This 6,742 page fully searchable biography of Meher Baba was written by Bhau Kalchuri shortly after Baba's death in 1969 and was published in a 20 volume set in 1986. It is based on journals kept by Meher Baba's followers from as early as 1922, as well as hundreds of hours of recorded interviews. 
* [http://discoursesbymeherbaba.org Discourses by Meher Baba] 
** The 1967 6th edition of Meher Baba's ''Discourses''. While some of these discourses were dictated verbatim by Meher Baba, others were worked up from points given by Baba to his close disciple Dr. C. D. Deshmukh, an Indian professor of philosophy. The discourses originally appeared in ''Meher Baba Journal'' from 1938 to 1943 and were later gathered as books, receiving numerous editions and revisions.
* [http://www.theawakenermagazine.org/ Awakener Magazine Archive] 
** ''Awakener Magazine'' was a magazine exclusively on Meher Baba that was published by Filis Frederick of California from 1953-1986. All 67 issues are now available online and are searchable. 
* [https://avatarmeherbabatrust.org/messages/]
** Messages given out by Meher Baba over the course of his life, often in the form of circulars or pamphlets.
* [http://www.avatarmeherbaba.org/erics/glossary.html Glossary of Meher Baba's terminology]
** An essential resource for studying the works of Meher Baba, this glossary defines terms as Meher Baba used them.


[[Category:Meher Baba Studies]]</text>
      <sha1>6oxcc9c3qrjo80rr0w8yp5tir0kdkrw</sha1>
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    <revision>
      <id>2816329</id>
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      <contributor>
        <username>Dazedbythebell</username>
        <id>243757</id>
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      <origin>2816329</origin>
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      <text bytes="2797" sha1="bsg5mx7mqyl1oljn7kleak1f06ox6s5" xml:space="preserve">[[Image:Meher_Baba_1957.jpg|thumb|Meher Baba]]
Welcome to the '''Center for the Study of Meher Baba''', part of the [[School:Theology|School of Theology]] and the [[Topic:Religious Studies|Division of Religious Studies]].

==Center Description==
The Wikiversity Center for the Study of [[w:Meher Baba|Meher Baba]] is a content development project where participants create, organize and develop Wikiversity content about Meher Baba.

== Learning Resources ==
* [[/Teachings and methodology/]]: an overview of Meher Baba's teachings and methods.

==External Learning Resources==

Fortunately there is a large body of source material by and about Meher Baba freely available on the internet. Many of Baba's major books can be searched on line or downloaded free in PDF format for research purposes. You are encouraged to make yourself familiar with what is freely available. Note that these materials are copyrighted by the Avatar Meher Baba Perpetual Public Charitable Trust. They are exclusively for research and educational purposes, and not for republication or resale. 

* [https://avatarmeherbabatrust.org/online-library-2/ The Trust Online Library] 
** ''The Trust Online Library'' has books by and about Meher Baba available to download as [[w:Portable Document Format|PDF]] files 
* [https://www.lordmeher.org/rev/index.jsp Lord Meher] 
** This 6,742 page fully searchable biography of Meher Baba was written by Bhau Kalchuri shortly after Baba's death in 1969 and was published in a 20 volume set in 1986. It is based on journals kept by Meher Baba's followers from as early as 1922, as well as hundreds of hours of recorded interviews. 
* [http://discoursesbymeherbaba.org Discourses by Meher Baba] 
** The 1967 6th edition of Meher Baba's ''Discourses''. While some of these discourses were dictated verbatim by Meher Baba, others were worked up from points given by Baba to his close disciple Dr. C. D. Deshmukh, an Indian professor of philosophy. The discourses originally appeared in ''Meher Baba Journal'' from 1938 to 1943 and were later gathered as books, receiving numerous editions and revisions.
* [http://www.theawakenermagazine.org/ Awakener Magazine Archive] 
** ''Awakener Magazine'' was a magazine exclusively on Meher Baba that was published by Filis Frederick of California from 1953-1986. All 67 issues are now available online and are searchable. 
* [https://avatarmeherbabatrust.org/messages/ Messages of Meher Baba]
** Messages given out by Meher Baba over the course of his life, often in the form of circulars or pamphlets.
* [http://www.avatarmeherbaba.org/erics/glossary.html Glossary of Meher Baba's terminology]
** An essential resource for studying the works of Meher Baba, this glossary defines terms as Meher Baba used them.


[[Category:Meher Baba Studies]]</text>
      <sha1>bsg5mx7mqyl1oljn7kleak1f06ox6s5</sha1>
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    <revision>
      <id>2816330</id>
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      <contributor>
        <username>Dazedbythebell</username>
        <id>243757</id>
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      <comment>Shorten description</comment>
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      <text bytes="2723" sha1="d96xaj0wb49svq84s6yzak87r6s3hks" xml:space="preserve">[[Image:Meher_Baba_1957.jpg|thumb|Meher Baba]]
Welcome to the '''Center for the Study of Meher Baba''', part of the [[School:Theology|School of Theology]] and the [[Topic:Religious Studies|Division of Religious Studies]].

==Center Description==
The Wikiversity Center for the Study of [[w:Meher Baba|Meher Baba]] is a content development project where participants create, organize and develop Wikiversity content about Meher Baba.

== Learning Resources ==
* [[/Teachings and methodology/]]: an overview of Meher Baba's teachings and methods.

==External Learning Resources==

Fortunately there is a large body of source material by and about Meher Baba freely available on the internet. Many of Baba's major books can be searched on line or downloaded free in PDF format for research purposes. You are encouraged to make yourself familiar with what is freely available. Note that these materials are copyrighted by the Avatar Meher Baba Perpetual Public Charitable Trust. They are exclusively for research and educational purposes, and not for republication or resale. 

* [https://avatarmeherbabatrust.org/online-library-2/ The Trust Online Library] 
** ''The Trust Online Library'' has books by and about Meher Baba available to download as [[w:Portable Document Format|PDF]] files 
* [https://www.lordmeher.org/rev/index.jsp Lord Meher] 
** This 6,742 page fully searchable biography of Meher Baba was written by Bhau Kalchuri shortly after Baba's death in 1969 and was published in a 20 volume set in 1986. It is based on journals kept by Meher Baba's followers from as early as 1922, as well as hundreds of hours of recorded interviews. 
* [http://discoursesbymeherbaba.org Discourses by Meher Baba] 
** The 1967 6th edition of Meher Baba's ''Discourses''. While some of these discourses were dictated verbatim by Meher Baba, others were worked up from points given by Baba to his close disciple Dr. C. D. Deshmukh, an Indian professor of philosophy. The discourses originally appeared in ''Meher Baba Journal'' from 1938 to 1943 and were later gathered as books, receiving numerous editions and revisions.
* [http://www.theawakenermagazine.org/ Awakener Magazine Archive] 
** ''Awakener Magazine'' was a magazine exclusively on Meher Baba that was published by Filis Frederick of California from 1953-1986. All 67 issues are now available online and are searchable. 
* [https://avatarmeherbabatrust.org/messages/ Messages of Meher Baba]
** Messages given out by Meher Baba over the course of his life, often in the form of circulars or pamphlets.
* [http://www.avatarmeherbaba.org/erics/glossary.html Glossary of Meher Baba's terminology]
** Defines terms as Meher Baba used them.


[[Category:Meher Baba Studies]]</text>
      <sha1>d96xaj0wb49svq84s6yzak87r6s3hks</sha1>
    </revision>
    <revision>
      <id>2816331</id>
      <parentid>2816330</parentid>
      <timestamp>2026-06-20T12:15:06Z</timestamp>
      <contributor>
        <username>Dazedbythebell</username>
        <id>243757</id>
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      <text bytes="2736" sha1="5q04j1s73n74apsxn426ai6rj411lfq" xml:space="preserve">[[Image:Meher_Baba_1957.jpg|thumb|Meher Baba, 1894–1969]]
Welcome to the '''Center for the Study of Meher Baba''', part of the [[School:Theology|School of Theology]] and the [[Topic:Religious Studies|Division of Religious Studies]].

==Center Description==
The Wikiversity Center for the Study of [[w:Meher Baba|Meher Baba]] is a content development project where participants create, organize and develop Wikiversity content about Meher Baba.

== Learning Resources ==
* [[/Teachings and methodology/]]: an overview of Meher Baba's teachings and methods.

==External Learning Resources==

Fortunately there is a large body of source material by and about Meher Baba freely available on the internet. Many of Baba's major books can be searched on line or downloaded free in PDF format for research purposes. You are encouraged to make yourself familiar with what is freely available. Note that these materials are copyrighted by the Avatar Meher Baba Perpetual Public Charitable Trust. They are exclusively for research and educational purposes, and not for republication or resale. 

* [https://avatarmeherbabatrust.org/online-library-2/ The Trust Online Library] 
** ''The Trust Online Library'' has books by and about Meher Baba available to download as [[w:Portable Document Format|PDF]] files 
* [https://www.lordmeher.org/rev/index.jsp Lord Meher] 
** This 6,742 page fully searchable biography of Meher Baba was written by Bhau Kalchuri shortly after Baba's death in 1969 and was published in a 20 volume set in 1986. It is based on journals kept by Meher Baba's followers from as early as 1922, as well as hundreds of hours of recorded interviews. 
* [http://discoursesbymeherbaba.org Discourses by Meher Baba] 
** The 1967 6th edition of Meher Baba's ''Discourses''. While some of these discourses were dictated verbatim by Meher Baba, others were worked up from points given by Baba to his close disciple Dr. C. D. Deshmukh, an Indian professor of philosophy. The discourses originally appeared in ''Meher Baba Journal'' from 1938 to 1943 and were later gathered as books, receiving numerous editions and revisions.
* [http://www.theawakenermagazine.org/ Awakener Magazine Archive] 
** ''Awakener Magazine'' was a magazine exclusively on Meher Baba that was published by Filis Frederick of California from 1953-1986. All 67 issues are now available online and are searchable. 
* [https://avatarmeherbabatrust.org/messages/ Messages of Meher Baba]
** Messages given out by Meher Baba over the course of his life, often in the form of circulars or pamphlets.
* [http://www.avatarmeherbaba.org/erics/glossary.html Glossary of Meher Baba's terminology]
** Defines terms as Meher Baba used them.


[[Category:Meher Baba Studies]]</text>
      <sha1>5q04j1s73n74apsxn426ai6rj411lfq</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Atcovi</title>
    <ns>3</ns>
    <id>106891</id>
    <revision>
      <id>2816375</id>
      <parentid>2814923</parentid>
      <timestamp>2026-06-21T03:21:50Z</timestamp>
      <contributor>
        <username>MediaWiki message delivery</username>
        <id>983498</id>
      </contributor>
      <comment>/* The Signpost: 21 June 2026 */ new section</comment>
      <origin>2816375</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10726" sha1="kox6s8q70u9mierr7u42e7omkw0rqnn" xml:space="preserve">[[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] • [[User talk:Atcovi/Archive 14 (April 15, 2023 - May 5, 2026)|/Archive 14 (April 15, 2023 - May 5, 2026)]]

:''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&amp;diff=750617&amp;oldid=740650 see this]'' 
{{tmbox
|small = 
|image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]]
|text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&amp;nbsp;{{{end}}}&amp;nbsp;}}{{#if:|due to&amp;nbsp;{{{reason}}}&amp;nbsp;}}and may not respond swiftly to queries.{{#if:|&lt;P&gt;{{{msg}}} }} 
| style     = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}}
}}

== Please vote ==

on Wikinews rebirth possibly on Wikiversity, thanks @[[User:Atcovi|Atcovi]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:21, 15 May 2026 (UTC)

:Hi BigKrow. I've been watching the discussion on the sidelines. Hopefully I'll have an input soon, I just have other commitments I'm catering to. Best of luck with your projects and welcome to Wikiversity! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:44, 16 May 2026 (UTC)

== ''The Signpost'': 22 May 2026 ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot; style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''&lt;/div&gt;
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* News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/News and notes|Offline: Osama Khalid still in prison]]

* In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In the media|Indonesian editors, you shall return!]]

* Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Disinformation report|Who is a typical paid editor? Who are their typical clients?]]

* Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Recent research|WikiLambda the Ultimate]]

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* Forum: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Forum|WikiAnnotate: help us build a dataset of article quality evaluations]]

* In focus: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In focus|Demystifying the 2026-27 Annual Plan]]

* Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Opinion|Wikipedia isn't a battleground. So why does it feel like one?]]

* Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Serendipity|Wikinews: Into the Wikiverse]]

* Special report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Special report|Wikimedia Foundation closes Wikinews after 21 years]]

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* Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Gallery|Earth Day and Mother's Day]]

* Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Comix|Brother, can you spare a page?]]

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== Wikiversity:Candidates for Bureaucratship/Atcovi ==

RE: [[Wikiversity:Candidates for Bureaucratship/Atcovi]] I have closed this as successful. Congrats! See [https://en.wikiversity.org/w/index.php?title=Wikiversity:Candidates_for_Bureaucratship/Atcovi&amp;diff=prev&amp;oldid=2812184] and [https://en.wikiversity.org/w/index.php?title=Special:Log&amp;logid=3549048]. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 19:23, 30 May 2026 (UTC)

:Thank you Mike! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:55, 30 May 2026 (UTC)
::Congratulations @[[User:Atcovi|Atcovi]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:58, 8 June 2026 (UTC)

== Question ==
Hello don't mean to bother and Ik its a silly question, on wikipedia there is a tool that allows for the creation of user boxes does wikiversity have it? Or should I create them myself like {{Userbox
| border-c = #000000
| id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]]
| id-c = #000000
| id-fc = #000000
| id-s = 14
| info = This user testified in front of the [[National Diet|national diet]]
| info-c = #000000
| info-fc = #ffffff
| info-s = 8
}}
however when i try to display them like on wikipedia i can't
{{yytop}}
{{yy|User:AUBSTRAWBS/GATE}}
{{yyend}}

Anyways sorry to bother you with somthing like this but i'm really stumped as to how to share them. Any help would be super apreciated also if you want any user boxes i can make them :). {{unsigned|AUBSTRAWBS}}
:Hello {{ping|AUBSTRAWBS}} no need to worry about bothering me, I'm always happy to help. I think for Wikiversity you'd have to manually create them, as I have done so. For example: [[Template:User Sri Lankan]] &amp; [[Template:User soccer]]. If you'd like to bring over templates from Wikipedia, then feel free to just copy them and paste them here - tho it may be better just to manually create them as it could be a lengthy and messy process. I do think the way you've created the &quot;national diet&quot; userbox is perfect and achieves the intended goal. Let me know if you have any more questions! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:38, 9 June 2026 (UTC)

:Here's one I created just now: [[Template:User university student]]. More templates are listed here: [[:Category:User templates]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:45, 9 June 2026 (UTC)
:Thank you very much [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 23:02, 9 June 2026 (UTC)

== ''The Signpost'': 21 June 2026 ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot; style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''&lt;/div&gt;
&lt;div style=&quot;column-count:2;&quot;&gt;
* From the editors: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/From the editors|Ways for beginners to support ''The Signpost'' community journalism]]

* News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/News and notes|Community Tech development team disbanded]]

* Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Disinformation report|PR for the people?]]

* Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Recent research|Proposed tagging system for AI involvement; successful and unsuccessful AI tools for contributors]]

* In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/In the media|Who won a 14th century battle and who won the 2026 Iran war?]]

* Community view: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Community view|Putting the Wish into the Wishlist]]

* In focus: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/In focus|A global standard for Neutral Point of View]]

* On the bright side: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/On the bright side|Flowers, blue helmets, reefs, pride, and Juneteenth]]

* Op-ed: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Op-ed|Breathe, Don’t Panic, there is a different story about Wikimedia + AI futures]]

* Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Opinion|Wikimedia Foundation staff develop union and Wikimedia user community reacts]]

* Technology report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Technology report|Community Tech team is disbanded, controversy erupts]]

* Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Traffic report|'Cause this is thriller, thriller night]]

* WikiConference report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/WikiConference report|Report of Volunteer Supporters Network Annual Meeting 2026]]

* Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Comix|Take your turn]]

* Humour: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Humour|Group of banned T-shirt makers comes out of hiding to sell new Wikipedia-themed merchandise]]

&lt;/div&gt;
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 03:21, 21 June 2026 (UTC)
&lt;!-- Sent via script ([[w:en:User:JPxG/SPS]]) --&gt;&lt;/div&gt;
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      <sha1>kox6s8q70u9mierr7u42e7omkw0rqnn</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2816334</id>
      <parentid>2816298</parentid>
      <timestamp>2026-06-20T14:15:56Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2816334</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3088" sha1="noj2b91wq911bs8jndie1lfggbtyu32" 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.2A.CLA.20260620.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260620.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>noj2b91wq911bs8jndie1lfggbtyu32</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2816339</id>
      <parentid>2816303</parentid>
      <timestamp>2026-06-20T14:25:43Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2816339</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="43bv4ryu3mo4wk4l89z611u70eqt745" 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.20260620.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>
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  </page>
  <page>
    <title>Universal Bibliography</title>
    <ns>0</ns>
    <id>171301</id>
    <revision>
      <id>2816370</id>
      <parentid>2803353</parentid>
      <timestamp>2026-06-21T02:31:35Z</timestamp>
      <contributor>
        <username>James500</username>
        <id>297601</id>
      </contributor>
      <comment>/* Languages */ Add</comment>
      <origin>2816370</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="46622" sha1="t3d7rzw3i6apua1j1sw5puyr2grb2dg" xml:space="preserve">{{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}}
{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; Dictionary of National Biography
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Languages==
Maltese
*See [[w:mt:Bibljografija tal-lingwa Maltija]]

Judaeo-Spanish (Ladino)
*See [[w:lad:Vikipedya:Bibliografia del djudeo-espanyol]]


Japonic
*Michinori Shimoji. An Introduction to the Japonic Languages: Grammatical Sketches of Japanese Dialects and Ryukyuan Languages. Brill. 2022. [https://books.google.co.uk/books?id=TO77EAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Yosuke Igarashi, Kenan Celik, Tatsuya Hirako and Hayato Aoi. Word-Prosodic Systems of Japonic Languages. Brill. 2026. [https://books.google.co.uk/books?id=B_3CEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Japan
*Masayoshi Shibatani. The Languages of Japan. CUP. 1990. [https://books.google.co.uk/books?id=sD-MFTUiPYgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series
*Handbooks of Japanese Language and Linguistics

Japanese
*Haruhiko Kindaichi. The Japanese Language. Tuttle. 1978. [https://books.google.co.uk/books?id=s_UZAQAAIAAJ]
1989. [https://books.google.co.uk/books?id=PdzkyasVMMoC]
*Osamu Mizutani. Japanese: The Spoken Language in Japanese Life. Japan Times. 1981. [https://books.google.co.uk/books?id=jZsPAAAAYAAJ]

Periodicals
*Japanese Language and Literature. (Journal of the Association of Teachers of Japanese.) [https://books.google.co.uk/books?&amp;id=QpkmAQAAIAAJ]

Introductions
*Richard Bowring and Haruko Uryū Laurie. An Introduction to Modern Japanese. 1992. [https://books.google.co.uk/books?id=Gu3k3eiOXWAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Understanding
*Yasuko Obana. Understanding Japanese: A Handbook for Learners and Teachers. 2000. [https://books.google.co.uk/books?id=I9IPAAAAYAAJ]

Learn
*Yuko Fukuroi. Learn Japanese. Institute of Asian Studies. 1997. [https://books.google.co.uk/books?id=0SJkAAAAMAAJ]
*John Young and Kimiko Nakajima-Okano. Learn Japanese: New College Text: Volume IV. 1985. [https://books.google.co.uk/books?id=rxwxLVwW2t0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Young and Kimiko Nakajima-Okano. Learn Japanese: Pattern Approach. University of Maryland. 1963. [https://books.google.co.uk/books?id=pG1AsovGf3AC]

*Miwa Kai. Listen &amp; Learn Japanese. 1959. Reprinted 1986. [https://books.google.co.uk/books?id=wBrYftZU6z4C&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Readings
*Joseph K Yamagiwa (ed). Readings in Japanese Language and Linguistics. University of Michigan Press. [https://books.google.co.uk/books?id=76wPAAAAYAAJ]

History
*Bjarke Frellesvig. A History of the Japanese Language. 2010. [https://books.google.co.uk/books?id=v1FcAgiAC9IC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lone Takeuchi. The Structure and History of Japanese: From Yamatokotoba to Nihongo. 1999. [https://books.google.co.uk/books?id=sr8PAAAAYAAJ]
*Ohno Susumu. The Origin of the Japanese Language. Kokusai Bunka Shinkokai. Tokyo. 1970. [https://books.google.co.uk/books?id=pqcPAAAAYAAJ]
*N A Syromiatnikov. The Ancient Japanese Language. Nauka Publishing House. 1981. [https://books.google.co.uk/books?id=OB5kAAAAMAAJ]

*Yaeko Sato Habein. The History of the Japanese Written Language. University of Tokyo Press. 1984. [https://books.google.co.uk/books?id=xh1kAAAAMAAJ]

Japanese and Ryukyuan
*Moriyo Shimabukuro. The Accentual History of the Japanese and Ryukyuan Languages: A Reconstruction. 2007. [https://books.google.co.uk/books?id=n_V5DwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ryukyuan
*Handbook of the Ryukyuan Languages: History, Structure, and Use [https://books.google.co.uk/books?id=g_FeCAAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Japanese and Korean
*J Marshall Unger. The Role of Contact in the Origins of the Japanese and Korean Languages. University of Hawaii Press. 2009. [https://books.google.co.uk/books?id=sYULAQAAMAAJ]

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Cinema==
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
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{{Bibliography}}
{{research}}

If this resource is ever completed, it will be a universal bibliography.&lt;ref&gt;See [[w:Bibliography]].&lt;/ref&gt; Until then, it will be an approximation of a universal bibliography.

This bibliography is arranged as an index of topics.

==Index==
*[[Universal Bibliography/Bibliography|Bibliography]]
*[[Universal Bibliography/Libraries|Libraries]]
*[[Universal Bibliography/Literature|Literature]]
*[[Universal Bibliography/SF|SF]]
*[[Universal Bibliography/Music|Music]]
*[[Universal Bibliography/Publishers and imprints|Publishers and imprints]]
*[[Universal Bibliography/Printing|Printing]]
*[[Universal Bibliography/Printers|Printers]]
*[[Universal Bibliography/Microform|Microform]]
*[[Universal Bibliography/Periodicals|Periodicals]]
*[[Universal Bibliography/Reference|Reference]]
*[[Universal Bibliography/Gazetteers|Gazetteers]]
*[[Universal Bibliography/Humanities|Humanities]]
*[[Universal Bibliography/Law|Law]]
*[[Universal Bibliography/History|History]]
*[[Universal Bibliography/Archaeology|Archaeology]]
*[[Universal Bibliography/Geography|Geography]]
*[[Universal Bibliography/Countries|Countries]]
*[[Universal Bibliography/Architecture|Architecture]]
*[[Universal Bibliography/Mathematics|Mathematics]]
*[[Universal Bibliography/Computers|Computers]]
*[[Universal Bibliography/Kites|Kites]]
*[[Universal Bibliography/Nostalgia|Nostalgia]]
*[[Universal Bibliography/Children's non-fiction|Children's non-fiction]]

===About===
*[[Universal Bibliography/About|About]]

==Online libraries==
Swedish:
*[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank)
*[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg)

==Biographical dictionaries etc==
See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]]

*Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Arthur, &quot;Biographical Dictionaries in the Digital Era&quot;. Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&amp;pg=PA83#v=onepage&amp;q&amp;f=false Page 83] et seq.

Bibliographies, indexes, etc:
*Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ]
*Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ]
*Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ]
*Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ]
*Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ]
*Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ]
*Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ]

Particular works:

*Oxford Dictionary of National Biography; Dictionary of National Biography
*Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&amp;btnG=Search+Books&amp;bksoutput=html_text&amp;tbm=bks&amp;tbo=1 editions:uzt3-qMuFcMC])
*A &amp; C Black's Who's Who
*Who Was Who
*The Academic Who's Who. A &amp; C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&amp;pg=PA208&amp;output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC
*Dictionary of Edwardian Biography (Pike's New Century Series)

Works that comprise largely of biographies:
*The Penguin Companion to Literature

Theatres
*A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers &amp; Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15]

==Dictionaries of dates==
[https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org]
*Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ]
*Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&amp;tbm=bks&amp;q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC]
*Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880
*[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ]
*Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440]
*Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930.
*[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]]
*Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates.
*Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will]
*Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&amp;pg=PA93&amp;output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA351&amp;output=html_text]
*Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May)
*Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&amp;id=k3dJAAAAYAAJ&amp;output=html]
*Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880]

Australia

*John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog]
*John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford &amp; Co. Brisbane. 1895.

America

*Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921.

==Commodity dictionaries==
*Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&amp;pg=PR22#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&amp;pg=RA15-PA7#v=onepage&amp;q&amp;f=false]
*Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, &quot;Commodity Dictionary&quot;, Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&amp;pg=PA182#v=onepage&amp;q&amp;f=false p 182] &amp; 183
*Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 &amp; 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ]
*Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC]
*Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&amp;pg=PA78#v=onepage&amp;q&amp;f=false]

==Encyclopedias==
See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]]

*Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990.
*The Dorling Kindersley Illustrated Family Encyclopedia

==Almanacs==
See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]].
*Year Book and Almanac of Newfoundland. 
**For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft]
*Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C]

==Censuses==
*Census of New Zealand and Labrador
**1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog]
**1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng]
**1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng]

==Pilot guides==
*[[w:United States Coast Pilot|United States Coast Pilot]]
*American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

*Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ]
*Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ]
*Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==Books of facts==
*The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC
*Chambers Book of Facts
*Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008
*Handy Book of Facts: Things Everyone Should Know. C.S. Hammond &amp; Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ]

==Series of books==
See [[w:Category:Series of books]] and [[w:Category:Monographic series]]
*George M Sinkankas, &quot;Series&quot; in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&amp;pg=PA250#v=onepage&amp;q&amp;f=false 250] to 273.
*&quot;Publishing in Series, 1896-1916&quot; in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&amp;pg=PA539#v=onepage&amp;q&amp;f=false Page 539] et seq.
*Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 2].
*Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ]
*Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Khan. &quot;Monographs in series&quot;. The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&amp;pg=PA208#v=onepage&amp;q&amp;f=false 207] to 209.
*Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&amp;pg=PA6#v=onepage&amp;q&amp;f=false 6] and 7.

*Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&amp;pg=PA11#v=onepage&amp;q&amp;f=false]
**Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&amp;pg=PA1190#v=onepage&amp;q&amp;f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&amp;pg=PA53#v=onepage&amp;q&amp;f=false]
***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ]
**Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ]
**Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ]
**Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ]
*Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ]
**2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ]
**3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA63#v=onepage&amp;q&amp;f=false]
*Ocran, Emmanuel Benjamin. Scientific &amp; Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&amp;pg=PA949#v=onepage&amp;q&amp;f=false]

*Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ]
*Young People's Literature in Series

*Catalog of Reprints in Series. (sometimes called &quot;Catalogue of Reprints in Series&quot;). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&amp;pg=PA73#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ]

*Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&amp;pg=PA656#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&amp;pg=PA367#v=onepage&amp;q&amp;f=false]

==Series of classics==
*Penguin Classics (Penguin Modern Classics, Penguin English Library)
*Oxford World Classics
*Everyman's Library
*Wordsworth Classics
*Macmillan Collectors Library
*Bantam Classics
*Minster Classics
*The Literary Heritage Collection (Heron Books, London. William Collins Sons &amp; Co, Glasgow)
*Chandos Classics
*Temple Classics
*Longmans Heritage of Literature Series

Russian
*Greatest Masterpieces of Russian Literature (Heron Books, London)

SF
*Corgi SF Collectors Library

Children's and shorter classics etc 
*Shorter Classics. Ginn and Company.
*Ladybird Children's Classics.
*Mini Classics. Parragon Books.
*Bonny Books. Peter Haddock Ltd.
*A series published by Dean &amp; Son Ltd

==Non-fiction general series==
*[[w:Oxford Companions|Oxford Companions]]
*[[w:Cambridge Companions|Cambridge Companions]]
*Princeton Companions
*Blackwell Companions. Wiley Blackwell Companions
*Routledge Companions. Routledge Research Companions
*Ashgate Companions. Ashgate Research Companions
*Brill's Companions
*Facts on File Companions
*Guides to Information Sources. Bowker-Saur
*Butterworths Guides to Information Sources.
*Columbia Guides
*Blackwell Guides
*Edinburgh Critical Guides
*Collins Reference Dictionaries
*New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]])
*Collins Gem (see [[w:List of Collins GEM books]])
*Concise Encyclopedias. Collins.
*Time Life Books (see [[w:Time Life#Book series]])
*[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. 
*[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton.
*Made Simple Books. W H Allen.
*Palgrave Master Series
*Harrap's Mini Series
*Shire Albums. Shire Publications.
*Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd.
*The Wonderful World Books. Macdonald and Company
*Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners.
*Hamlyn Pocket Guides
*Oxford Monograph Series
*Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]]
*Helpmate Handbooks. Willow Books

University
*University Paperbacks. Meuthen &amp; Co
*World Student Series. Addison Wesley
*Unibooks. Hodder and Stoughton
*International Student Editions. Van Nostrand Reinhold
*Hutchinson University Library

Imprints
*Pelican Books

Pictorials
*Salmon Cameracolour series
*Pitkin Pictorials 

United Kingdom
*Aspects of Britain. HMSO.

Places
*The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]]
*G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]]

Art
*Movements in World Art. Meuthen.
*Movements in Modern Art. Meuthen.
*How to Draw and Paint. New Burlington.

Film
*BFI Companions

Popular science
*Contemporary Science Paperbacks. Oliver and Boyd.
*Pan Piper Science Series

Science and mathematics
*Simon and Schuster Tech Outlines
*Schaum's Outline Series

Military
*Illustrated Military Guides. Illustrated Guides. &quot;An Illustrated Guide to ...&quot;. Salamander Books.
*Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press.
*Osprey Men-at-Arms
*Jane's Pocket Books

Communication
*The Library of Communication Techniques. Focal Press.
*John Fiske (ed). Studies in Culture and Communication. Routledge.
*The Media. Wayland.

Cookery
*ABC series. Peter Pauper Press.

Gardening
*Pan Piper Small Gardens Series.

Mythology
*Series on mythology published by Southwater (imprint of Anness)

==History and Geography==
See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]].
*Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

General series
*Cambridge Studies in Historical Geography

Atlases
*The Times Atlas of World History
*Philip's Atlas of World History

History of geography:
*Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ]

==Chronology==
See also [[Universal Bibliography/History#Millennia, centuries and decades]]

General
*Chronology of World History.
**Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994.
**Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994.
**Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994.
**Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon &amp; Schuster. Reissued 1994.

Centuries
*Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ]
*Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ]
*Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC]
*20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ]
*Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ]
*Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Decades
*Series:
**Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ]

Years
*Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&amp;pg=PA113#v=onepage&amp;q&amp;f=false Catalogue].
*The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false 1882 to 1883].
*James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false 1821].
*Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ancient
*Bickerman. Chronology of the Ancient World. 1968.
*Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993.

==Anniversaries==
*Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ]
*On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

==Egyptology==
*Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

==Battlefields==
*[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series]
*&quot;The Times Guide to Battlefields of Britain&quot;. Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart.

==Armed forces==
Periodicals:
*[[w:NATO Review|NATO Review]]

Military
*The Journal of Military History

*Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC]

*The Monthly Army List [Google editions:I0t2L4ElznEC]
*The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC]

*Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC]

*The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&amp;q=editions:dEauCcI7kssC&amp;biw=534&amp;bih=736&amp;dpr=1.5#sbfbu=1]
*The Royal Tank Corps Journal
*The Tank [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:Dv-RbpoM7acC&amp;biw=534&amp;bih=736&amp;dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment
*The Cavalry Journal [https://www.google.com/search?sa=N&amp;cs=0&amp;tbm=bks&amp;q=editions:cVQlfkRl6KUC&amp;biw=534&amp;bih=688&amp;dpr=1.5#sbfbu=1]
*The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&amp;q=editions:liFy4uc0ggYC&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC]
*The Royal Engineers Journal [https://www.google.com/search?tbm=bks&amp;q=editions:8XobinXLbD0C&amp;biw=534&amp;bih=736&amp;dpr=1.5]
*Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C]
*Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC]

United States
*Military Review
*The Coast Artillery Journal [Google editions:nMCogSJ_rlkC]
*Infantry Journal [Google editions:ULqoLmbUR5cC]
*The Reserve Officer [Google editions:JQDRDrnD1QQC]

Naval
*[[w:Navy News|Navy News]]

==Armour==
Armoured warfare; tank warfare
*Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ]
*Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ]
*Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ]
*Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ]
*Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ]
*Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ]

Atlas
*Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ]

Armored forces
*Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ]

==Mesoamerica==
*James. Aztecs &amp; Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: &quot;Books Received&quot; (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and &quot;Overviews for the general reader&quot; (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252.
*Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ]

==Accounting==
See [[s:Category:Accounting]]

Periodicals
*[[s:The Accountant|The Accountant]] (1874 onwards)
*Accountant's Magazine (1897 onwards) Aberdeen

==Arts==
*Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997.

==Biography==
*Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ]
*Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ]

Yearbooks
*Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC]
*Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ]

==Information technology==
*Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

==Economics==
General series:
*Dryden Press Series in Economics
*Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational
*Nuffield Economics &amp; Business. Nuffield Foundation. Longman.

Other:
*Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984.

*Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991.
*Anderton, Alain. Economics. Causeway Press. 1991.
*Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983.
*Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991.
*Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986.

*Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989.

*Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985. 

*Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981.

*National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990.
*Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992.

==Games==
Chess
*Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987.
*Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980.
*Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977.
*Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. 

General series
*Batsford Chess Books
**Discovering Chess Series. B T Batsford.

Periodicals

See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]]
*British Chess Magazine

Wargames
*Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward.
**Laurie Taylor. &quot;Attila the Hun invades Tyne Tees&quot;. TV Times. 1978. pp 28 &amp; 29.
**Terry Wise. &quot;Battleground&quot;. Battle for Wargamers. June 1978. pp 261 &amp; 262.
*[[w:Game of War|Game of War]]. Channel 4. 1997. [Television].

==Cricket==
See [[w:Bibliography of cricket]]

*Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ].
**Peter Arnold. The Illustrated Encyclopedia of World Cricket.
*Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ]

Scores and biographies
*Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]]

Periodicals
*[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false].

Australia
*Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463]
*The Oxford Companion to Australian Cricket

India 
*The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies]

Annuals
*[[w:Indian Cricket (annual)|Indian Cricket]].  [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966].

===Cricketers===
Cricketers, including biographical dictionaries and collections of biographies
*[[w:ESPNcricinfo|ESPNcricinfo]]
*[[w:CricketArchive|CricketArchive]]

*John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ]

*World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ]
*Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ]

*Sproat. Debrett's Cricketers' Who's Who. 1980.
*S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934.
*Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ]

*Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ]

*Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC]
*Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ]

Gloucestershire
*Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html]
*Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C].
*Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795]

Australia
*The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ]
*Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ]
*C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894.
*Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus &amp; Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ].

==Geology==
*Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History.

==Mineralogy==
*Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&amp;pg=PA473#v=onepage&amp;q&amp;f=false 473] to 476.

*Battey, Maurice Hugh. Mineralogy for students. Oliver &amp; Boyd. 1972. 2nd Ed. Longman. 1981.

==Paper==
See [[s:Category:Paper]]

*Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf]
*West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog]

==Books==
*British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ]
*Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ]

*Book News. 1882 to 1918. (John Wanamaker). Called &quot;Book News Monthly&quot; from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ]
*U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ]
*Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ]

*Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ]
*Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C]

*Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ]

Rare
*Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ]

Printed
*Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&amp;pg=PP1#v=onepage&amp;q&amp;f=false vol 31]

==Paperback and Paperbound==
*Swados, &quot;Paper Books: What do they Promise?&quot; (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114
*Wagman, &quot;The Paperbound Book Business&quot; (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&amp;pg=RA4-PA9#v=onepage&amp;q&amp;f=false 9] (No 5, November)

==Languages==
Maltese
*See [[w:mt:Bibljografija tal-lingwa Maltija]]

Judaeo-Spanish (Ladino)
*See [[w:lad:Vikipedya:Bibliografia del djudeo-espanyol]]


Japonic
*Michinori Shimoji. An Introduction to the Japonic Languages: Grammatical Sketches of Japanese Dialects and Ryukyuan Languages. Brill. 2022. [https://books.google.co.uk/books?id=TO77EAAAQBAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Yosuke Igarashi, Kenan Celik, Tatsuya Hirako and Hayato Aoi. Word-Prosodic Systems of Japonic Languages. Brill. 2026. [https://books.google.co.uk/books?id=B_3CEQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Japan
*Masayoshi Shibatani. The Languages of Japan. CUP. 1990. [https://books.google.co.uk/books?id=sD-MFTUiPYgC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Series
*Handbooks of Japanese Language and Linguistics

Japanese
*Haruhiko Kindaichi. The Japanese Language. Tuttle. 1978. [https://books.google.co.uk/books?id=s_UZAQAAIAAJ]
1989. [https://books.google.co.uk/books?id=PdzkyasVMMoC]
*Osamu Mizutani. Japanese: The Spoken Language in Japanese Life. Japan Times. 1981. [https://books.google.co.uk/books?id=jZsPAAAAYAAJ]

Periodicals
*Japanese Language and Literature. (Journal of the Association of Teachers of Japanese.) [https://books.google.co.uk/books?&amp;id=QpkmAQAAIAAJ]

Introductions
*Richard Bowring and Haruko Uryū Laurie. An Introduction to Modern Japanese. 1992. [https://books.google.co.uk/books?id=Gu3k3eiOXWAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Understanding
*Yasuko Obana. Understanding Japanese: A Handbook for Learners and Teachers. 2000. [https://books.google.co.uk/books?id=I9IPAAAAYAAJ]

Learn
*Yuko Fukuroi. Learn Japanese. Institute of Asian Studies. 1997. [https://books.google.co.uk/books?id=0SJkAAAAMAAJ]
*John Young and Kimiko Nakajima-Okano. Learn Japanese: New College Text: Volume IV. 1985. [https://books.google.co.uk/books?id=rxwxLVwW2t0C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*John Young and Kimiko Nakajima-Okano. Learn Japanese: Pattern Approach. University of Maryland. 1963. [https://books.google.co.uk/books?id=pG1AsovGf3AC]
*Nobuko Mizutani. Let's Learn Japanese. (Radio Japan). 1993. [https://books.google.co.uk/books?id=4urrPQAACAAJ]
*Senko K Maynard. Learning Japanese for Real: A Guide to Grammar, Use, and Genres of the Nihongo World. University of Hawaii Press. 2011. [https://books.google.co.uk/books?id=QF4EEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Miwa Kai. Listen &amp; Learn Japanese. 1959. Reprinted 1986. [https://books.google.co.uk/books?id=wBrYftZU6z4C&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Readings
*Joseph K Yamagiwa (ed). Readings in Japanese Language and Linguistics. University of Michigan Press. [https://books.google.co.uk/books?id=76wPAAAAYAAJ]

History
*Bjarke Frellesvig. A History of the Japanese Language. 2010. [https://books.google.co.uk/books?id=v1FcAgiAC9IC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Lone Takeuchi. The Structure and History of Japanese: From Yamatokotoba to Nihongo. 1999. [https://books.google.co.uk/books?id=sr8PAAAAYAAJ]
*Ohno Susumu. The Origin of the Japanese Language. Kokusai Bunka Shinkokai. Tokyo. 1970. [https://books.google.co.uk/books?id=pqcPAAAAYAAJ]
*N A Syromiatnikov. The Ancient Japanese Language. Nauka Publishing House. 1981. [https://books.google.co.uk/books?id=OB5kAAAAMAAJ]

*Yaeko Sato Habein. The History of the Japanese Written Language. University of Tokyo Press. 1984. [https://books.google.co.uk/books?id=xh1kAAAAMAAJ]

Japanese and Ryukyuan
*Moriyo Shimabukuro. The Accentual History of the Japanese and Ryukyuan Languages: A Reconstruction. 2007. [https://books.google.co.uk/books?id=n_V5DwAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Ryukyuan
*Handbook of the Ryukyuan Languages: History, Structure, and Use [https://books.google.co.uk/books?id=g_FeCAAAQBAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Japanese and Korean
*J Marshall Unger. The Role of Contact in the Origins of the Japanese and Korean Languages. University of Hawaii Press. 2009. [https://books.google.co.uk/books?id=sYULAQAAMAAJ]

==Science==
*Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998.

==Entertainment==
*The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/]

==Television==
*Rob Young. The Magic Box: Viewing Britain Through the Rectangular Window. [https://books.google.co.uk/books?id=fH8NEAAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]. Review: [https://www.theguardian.com/books/2021/aug/13/the-magic-box-by-rob-young-review-a-spirited-history-of-television]

Magazines
*The Radio Times
*TV Times

Newspaper television reviews etc

United Kingdom
*A A Gill. Paper View: The Best of the Sunday Times Television Columns.

*&quot;Choice&quot; or &quot;Television and Radio Choice&quot; in &quot;Television and Radio&quot;. 1991.  Middle of newspaper. The page number of the listings is given on the front page. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992. These reviews are printed in the body of the listings, and not in a separate column. These reviews are printed on the last page of the &quot;Life &amp; Times&quot; section of the newspaper, for issues of the newspaper where &quot;Life &amp; Times&quot; is a separate section. Otherwise they are printed in the middle of newspaper.
*&quot;Choice&quot; or &quot;TV Choice&quot; in &quot;Television and Radio&quot;. The Times. 1992 to 1993. Penultimate page of newspaper. These reviews are printed in the body of the listings, and not in a separate column.
*&quot;Choice&quot;. The Times. 1993 to 1997. Mondays to Fridays. Penultimate page of newspaper.
*&quot;Television Choice&quot;. The Times. 1997 onwards. Mondays to Fridays. Third page from back of newspaper.
*&quot;Review&quot;. The Times. 1994 onwards. Mondays to Fridays. Penultimate page of newspaper.
*There are reviews in:
**The Independent, The Guardian, The Financial Times, and The Daily Telegraph

Netherlands
*&quot;TV: Films Video&quot; in &quot;televisie en radio woensdag&quot;. Limburgs Dagblad.
*&quot;show&quot;. Limburgs Dagblad.

Japan
*&quot;Today's Choice&quot; in &quot;TV/Radio&quot;. The Japan Times.

Music
*Tele-Tunes

Archives and listings
*[https://www.nhk.or.jp/archives/ NHK Archives]. [https://www.nhk.or.jp/archives/chronicle/ Chronicle]. [https://www.nhk.or.jp/archives/chronicle/timetable/ Timetables].

==Cinema==
*Edgar Anstey, &quot;The Cinema&quot; (1944) 172 The Spectator 10 (No 6028: 7 January 1944). Includes &quot;Review of the Year&quot;.

==Animation==
*John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005.
*Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&amp;pg=PA1958#v=onepage&amp;q&amp;f=false].

==Colours==
*Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/]

==Culture==
*Eagleton. Culture. 2016. [https://books.google.co.uk/books?id=z2EdDAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Highmore. Culture. 2016. [https://books.google.co.uk/books?id=2teoCgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Jenks. Culture. 1993. [https://books.google.co.uk/books?id=6Litru5-ImAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

*Crane. The Production of Culture. 1992. [https://books.google.co.uk/books?id=DGs5DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Calhoun and Sennett. Practicing Culture. 2007. [https://books.google.co.uk/books?id=NbO4CDIWhn4C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Mead. The Study of Culture at a Distance. 1953. 2000. [https://books.google.co.uk/books?id=5Upv9RZfPe8C&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Measuring Culture. 2020. [https://books.google.co.uk/books?id=0se_DwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Popular culture
*Kornhaber. [https://www.theatlantic.com/magazine/archive/2025/06/american-pop-culture-decline/682578/ Is This the Worst-Ever Era of American Pop Culture?]. The Atlantic. 5 May 2025. (June 2025 issue).

==Bilateral==
Britain and Japan
*Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ]

==Prehistoric life==
Prehistoric animals
*[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.]
*Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&amp;pg=PA269#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&amp;pg=PA5#v=onepage&amp;q&amp;f=false].
**Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977.  Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ]
**Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme.

Dinosaurs
*Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.]

==Continents==
===Asia===
====Far East====
Bibliography
*Kuniyoshi. Far East. (PACAF Basic Bibliographies). 1957. [https://books.google.co.uk/books?id=Q5TLdCbP2HcC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

==See also==
*[[Bibliography]]

==Notes==
{{Reflist}}
{{subpagesif}}
[[Category:Bibliographies]]
[[Category:Research]]</text>
      <sha1>6h5vbwr39oezb40on4fpv2rki232jlv</sha1>
    </revision>
  </page>
  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
    <id>212733</id>
    <revision>
      <id>2816359</id>
      <parentid>2815515</parentid>
      <timestamp>2026-06-21T00:40:25Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2816359</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="lftb39qoo9g84miojlntnlz9wr7j1ae" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260615.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>lftb39qoo9g84miojlntnlz9wr7j1ae</sha1>
    </revision>
    <revision>
      <id>2816361</id>
      <parentid>2816359</parentid>
      <timestamp>2026-06-21T00:41:35Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2816361</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="tie13dcl8k6m5wme3wwpah7s3ztp23s" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260616.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>tie13dcl8k6m5wme3wwpah7s3ztp23s</sha1>
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    <revision>
      <id>2816363</id>
      <parentid>2816361</parentid>
      <timestamp>2026-06-21T00:42:27Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2816363</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="exmg5hdbukq0n6cm6r1j3n88lk4zzvw" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260617.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>exmg5hdbukq0n6cm6r1j3n88lk4zzvw</sha1>
    </revision>
    <revision>
      <id>2816365</id>
      <parentid>2816363</parentid>
      <timestamp>2026-06-21T00:43:24Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2816365</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="m0i732tmj1vb0ntlmawq5vyyrtctydz" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260618.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>m0i732tmj1vb0ntlmawq5vyyrtctydz</sha1>
    </revision>
  </page>
  <page>
    <title>Research in programming Wikidata/Business enterprise</title>
    <ns>0</ns>
    <id>223822</id>
    <revision>
      <id>2816385</id>
      <parentid>2816317</parentid>
      <timestamp>2026-06-21T09:07:58Z</timestamp>
      <contributor>
        <username>Ratte</username>
        <id>1445482</id>
      </contributor>
      <minor />
      <comment>/* Test */</comment>
      <origin>2816385</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21851" sha1="58rgm6ixydo8y6w9hrf0xy49x1xtoyr" xml:space="preserve">This article is devoted to the study Wikidata objects &quot;commercial organizations&quot;. With the help of SPARQL queries, computed on the objects of the type &quot;commercial organizations&quot; in the Wikidata, the following tasks have been solved: maked a list with organizations by branches distribution in the form of a bubble chart, counted the quantity of organizations by countries, drawn the graph of existing organizations and their subsidiaries. Conclusions were drawn regarding the completeness of the Wikidata on this topic, including a map of the organizations of the world.

== Instances of object &quot;Business enterprise&quot; ==
* Objects: [[d:Q4830453|business enterprise (Q4830453)]]

Using the following queary we can get list of all commercial organizations.

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;#added 2017-02
#List of `instances of` &quot;business enterprise&quot; 
SELECT ?lang ?langLabel
WHERE
{
    ?lang wdt:P31 wd:Q4830453.
    SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot; }
}
&lt;/syntaxhighlight&gt;
[https://query.wikidata.org/#%23List%20of%20%60instances%20of%60%20%22business%20enterprise%22%20%0ASELECT%20%3Flang%20%3FlangLabel%0AWHERE%0A%7B%0A%20%20%20%20%3Flang%20wdt%3AP31%20wd%3AQ4830453.%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%7D SPARQL-query], 109383 Results 

&lt;span style=&quot;color:green&quot;&gt;👍&lt;/span&gt;&gt; The most complete and elaborated business enterprise on the Wikidata are: [[d:Q95|Google]], [[d:Q312|Apple]], 
[[d:Q2283|Microsoft]]

&lt;span style=&quot;color:red&quot;&gt;👎&lt;/span&gt;&gt; Almost empty and uninformative business enterprise on the Wikidata are: [[d:Q40987|Pininfarina]], [[d:Q46065|ANHUI EXPRESSWAY COMPANY LIMITED]], [[d:Q45812|Futura et Marge]]

The defect of the resulting list is that objects turned out to be nameless on the Wikidata (No label defined). Let's try to get a list of organizations where &quot;label&quot; field will be non-empty.

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;#List of `instances of` &quot;business enterprise&quot; only with a label.
SELECT ?item ?item_label
WHERE
{
    ?item wdt:P31 wd:Q4830453
    ; rdfs:label ?item_label. 

    FILTER (LANG(?item_label) = &quot;en&quot;). 
}
&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#SELECT%20%3Fitem%20%3Fitem_label%0AWHERE%0A%7B%0A%20%20%20%20%3Fitem%20wdt%3AP31%20wd%3AQ4830453%0A%20%20%20%20%3B%20rdfs%3Alabel%20%3Fitem_label%20.%20%0A%0A%20%20%20%20FILTER%20%28LANG%28%3Fitem_label%29%20%3D%20%22en%22%29%20.%20%0A%7D SPARQL-query], 74556 Results

== Distribution of organizations by industry ==
Each organization specializes some industry. In order to understand which industry, for example, is the most popular (that is, how many organizations work in this industry), we can build a diagram.

Type of result: bubble diagram.

Are used:
* object [[d:Q4830453|business enterprise (Q4830453)]] (business enterprise),
* property [[d:Property:P452|industry (P452)]] (industry).

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#enterprise industry ranking
#defaultView:BubbleChart
SELECT ?industry ?company (count(*) as ?count)
WHERE 
{
    ?org wdt:P31 wd:Q4830453.
    ?org wdt:P452 ?industry.
    OPTIONAL {
		?industry rdfs:label ?company
		filter (lang(?company) = &quot;en&quot;)
	}
}
GROUP BY ?industry ?company
ORDER BY DESC(?count) ASC(?company)
&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23enterprise%20industry%20ranking%0A%23defaultView%3ABubbleChart%0ASELECT%20%3Findustry%20%3Fcompany%20%28count%28%2a%29%20as%20%3Fcount%29%0AWHERE%20%0A%7B%0A%20%20%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%0A%20%20%20%20%3Forg%20wdt%3AP452%20%3Findustry%20.%0A%20%20%20%20OPTIONAL%20%7B%0A%09%09%3Findustry%20rdfs%3Alabel%20%3Fcompany%0A%09%09filter%20%28lang%28%3Fcompany%29%20%3D%20%22en%22%29%0A%09%7D%0A%7D%0AGROUP%20BY%20%3Findustry%20%3Fcompany%0AORDER%20BY%20DESC%28%3Fcount%29%20ASC%28%3Fcompany%29%0A SPARQL query], 864 Results.

After analysis of this diagram (Fig. 1), we can conclude that the number of organizations involved in a particular industry. It is possible to build a table based on the data obtained (make a list of the 5 most popular industries):

&lt;table border=&quot;1&quot;&gt;
   &lt;caption&gt;TOP5 most popular industries&lt;/caption&gt;
   &lt;tr&gt;
    &lt;th&gt;Industry name&lt;/th&gt;
    &lt;th&gt;Quantity of organizations&lt;/th&gt;
   &lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;automative industry&lt;/td&gt;&lt;td&gt;1149&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;retail&lt;/td&gt;&lt;td&gt;843&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;telecommunications&lt;/td&gt;&lt;td&gt;648&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;video game industry&lt;/td&gt;&lt;td&gt;633&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;manufacturing&lt;/td&gt;&lt;td&gt;506&lt;/td&gt;&lt;/tr&gt;
&lt;/table&gt;

[[File:Diagram of organizations of the world by industry.jpg|thumb|Fig. 1: Diagram of organizations of the world by industry|center|900px]]&lt;br style=&quot;clear: both;&quot;&gt;

Let's answer the question: What and how many industries exist in Russia?

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#enterprise industry ranking in Russia
#defaultView:BubbleChart
SELECT ?industry ?company (count(*) as ?count) 
WHERE 
{
    ?org wdt:P31 wd:Q4830453.
    ?org wdt:P452 ?industry.
    ?org wdt:P17 wd:Q159. #Russia country
    OPTIONAL {
		?industry rdfs:label ?company
		filter (lang(?company) = &quot;en&quot;)
	}
}
GROUP BY ?industry ?company
ORDER BY DESC(?count) ASC(?company)
&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23enterprise%20industry%20ranking%0A%23defaultView%3ABubbleChart%0ASELECT%20%3Findustry%20%3Fcompany%20%28count%28%2a%29%20as%20%3Fcount%29%20%0AWHERE%20%0A%7B%0A%20%20%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%0A%20%20%20%20%3Forg%20wdt%3AP452%20%3Findustry%20.%0A%20%20%20%20%3Forg%20wdt%3AP17%20wd%3AQ159.%20%23Russia%20country%0A%20%20%20%20OPTIONAL%20%7B%0A%09%09%3Findustry%20rdfs%3Alabel%20%3Fcompany%0A%09%09filter%20%28lang%28%3Fcompany%29%20%3D%20%22en%22%29%0A%09%7D%0A%7D%0AGROUP%20BY%20%3Findustry%20%3Fcompany%0AORDER%20BY%20DESC%28%3Fcount%29%20ASC%28%3Fcompany%29%0A SPARQL-query], 60 Results.

&lt;table border=&quot;1&quot;&gt;
   &lt;caption&gt;TOP5 most popular organizations in Russia&lt;/caption&gt;
   &lt;tr&gt;
    &lt;th&gt;Industry name&lt;/th&gt;
    &lt;th&gt;Quantity of organizations&lt;/th&gt;
   &lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;retail&lt;/td&gt;&lt;td&gt;78&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;automative industry&lt;/td&gt;&lt;td&gt;13&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;arms industry&lt;/td&gt;&lt;td&gt;10&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;aerospace industry&lt;/td&gt;&lt;td&gt;9&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;video game industry&lt;/td&gt;&lt;td&gt;9&lt;/td&gt;&lt;/tr&gt;
&lt;/table&gt;

It can be concluded that such industry as retail in Russia dominates over the rest, and very seriously. If the quantity of organizations in this area reaches 78, then in the next industry (automotive industry), only 13 organizations work.

For comparison, we can build a list of existing industries of some other country (for example, Norway).

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#enterprise industry ranking in Norway
#defaultView:BubbleChart
SELECT ?industry ?company (count(*) as ?count) 
WHERE 
{
    ?org wdt:P31 wd:Q4830453.
    ?org wdt:P452 ?industry.
    ?org wdt:P17 wd:Q20. #Norway country
    OPTIONAL {
		?industry rdfs:label ?company
		filter (lang(?company) = &quot;en&quot;)
	}
}
GROUP BY ?industry ?company
ORDER BY DESC(?count) ASC(?company)
&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23enterprise%20industry%20ranking%20in%20Norway%0A%23defaultView%3ABubbleChart%0ASELECT%20%3Findustry%20%3Fcompany%20%28count%28%2a%29%20as%20%3Fcount%29%20%0AWHERE%20%0A%7B%0A%20%20%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%0A%20%20%20%20%3Forg%20wdt%3AP452%20%3Findustry%20.%0A%20%20%20%20%3Forg%20wdt%3AP17%20wd%3AQ20.%20%23Norway%20country%0A%20%20%20%20OPTIONAL%20%7B%0A%09%09%3Findustry%20rdfs%3Alabel%20%3Fcompany%0A%09%09filter%20%28lang%28%3Fcompany%29%20%3D%20%22en%22%29%0A%09%7D%0A%7D%0AGROUP%20BY%20%3Findustry%20%3Fcompany%0AORDER%20BY%20DESC%28%3Fcount%29%20ASC%28%3Fcompany%29%0A SPARQL-query], 41 Results.

The dominant industry here is [[d:Q187939|manufacturing (Q187939)]].

== Number of organizations by country ==

Next query displays number of commercial organizations in each country in the world.

Are used:
* object [[d:Q4830453|business enterprise (Q4830453)]] (business enterprise),
* property [[d:Property:P17|country (P17)]] (country).

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
SELECT ?countryLabel (count(?org) as ?count)
WHERE
{
    ?org  wdt:P31 wd:Q4830453.
    ?org wdt:P17 ?country.

  SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot; }
 }
  GROUP BY ?country ?countryLabel
  ORDER BY DESC (?count)
&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#SELECT%20%3FcountryLabel%20%28count%28%3Forg%29%20as%20%3Fcount%29%0AWHERE%0A%7B%0A%20%20%20%20%3Forg%20%20wdt%3AP31%20wd%3AQ4830453.%0A%20%20%20%20%3Forg%20wdt%3AP17%20%3Fcountry.%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%20%7D%0A%20%20GROUP%20BY%20%3Fcountry%20%3FcountryLabel%0A%20%20ORDER%20BY%20DESC%20%28%3Fcount%29%0A SPARQL-query], 198 Results

== Organizations and their subsidiaries ==

It is necessary to build a graph from existing organizations, including subsidiaries.

Are used:
* object [[d:Q4830453|business enterprise (Q4830453)]] (business enterprise),
* property [[d:Property:P355|subsidiary (P355)]] (subsidiary).

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#subsidary graph
#defaultView:Graph
SELECT ?org ?orgLabel ?subsidiary ?subsidiaryLabel
WHERE
{
    ?org wdt:P31 wd:Q22687
    ; rdfs:label ?item_label.

    SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot; }
    OPTIONAL { ?org wdt:P355 ?subsidiary. }
    FILTER  (LANG(?item_label) = &quot;en&quot;) 
}
&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23neighboring%20countries%20graph%0A%23defaultView%3AGraph%0ASELECT%20%3Forg%20%3ForgLabel%20%3Fsubsidary%20%3FsubsidaryLabel%0AWHERE%0A%7B%0A%20%20%20%20%3Forg%20wdt%3AP31%20wd%3AQ22687%0A%20%20%20%20%3B%20rdfs%3Alabel%20%3Fitem_label%20.%0A%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%20%20%20%20OPTIONAL%20%7B%20%3Forg%20wdt%3AP355%20%3Fsubsidary%20.%20%7D%0A%20%20%20%20FILTER%20%20%28LANG%28%3Fitem_label%29%20%3D%20%22en%22%29%20%0A%7D%0A SPARQL-query], 428 Results(edges).

The resulting graph of neighbors (Fig. 2) consists of hanging vertices and isolated vertices. It is necessary to construct a graph where these vertices are absent.

[[File:Diagram of subsidiaries of the world.jpg|thumb|Fig. 2: Diagram of subsidiaries of the world|center|900px]]&lt;br style=&quot;clear: both;&quot;&gt;

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#neighboring countries graph
#defaultView:Graph
SELECT ?org ?orgLabel ?subsidiary ?subsidiaryLabel
WHERE
{
    ?org wdt:P31 wd:Q22687
    ; rdfs:label ?item_label.
    ?org wdt:P355 ?subsidiary. 
  
    SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot; }

    FILTER  (LANG(?item_label) = &quot;en&quot;) 
}
&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23neighboring%20countries%20graph%0A%23defaultView%3AGraph%0ASELECT%20%3Forg%20%3ForgLabel%20%3Fsubsidary%20%3FsubsidaryLabel%0AWHERE%0A%7B%0A%20%20%20%20%3Forg%20wdt%3AP31%20wd%3AQ22687%0A%20%20%20%20%3B%20rdfs%3Alabel%20%3Fitem_label%20.%0A%20%20%20%20%3Forg%20wdt%3AP355%20%3Fsubsidary%20.%20%0A%20%20%0A%20%20%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%20%7D%0A%0A%20%20%20%20FILTER%20%20%28LANG%28%3Fitem_label%29%20%3D%20%22en%22%29%20%0A%7D%0A SPARQL-query], 55 Results(edges).

== Fullness of the Wikidata ==

According to the category [[w: en: List_of_companies_of_Russia | List of companies of Russia]] there are at least 208 commercial organizations in English Wikipedia in Russia. We can note that there is a rating of the largest companies of Russia that is listed. It can be concluded that even big organizations have not been included in this list, not talking about small and medium ones.

It is impossible to obtain relevant data on the number of commercial organizations, because their number grows every day, and information about them is not represented in the public domain. For example, the USRLE, which provides data for a fee. {{Sfn|EGRUL|2017}}

The quantity of commercial organizations entered in the state register as newly created, in 2014 amounted 420.5 thousand, according to data on the site of the Federal Tax Service (FTS). In June, 2015 came into force orders of the Ministry of Finance of Russia that the data of existing organizations and information about them no longer applies in public. The data can be provided only to state authorities, local self-government bodies and so on. Therefore, it is not possible to obtain reliable data on the quantity of available organizations.

There is an opportunity to explore fullness with the help of the Wikidata. It is necessary to remember the total number of organizations (from the beginning) on the Wikidata (about 110 000, as their number is constantly growing). A typical user who has a general understanding of organizations may be interested to see how an organization looks or where it is located on the map.

To see how many organizations have an image (that is, the 'image' field is filled in), we need to write the following script.

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#List of organizations with image

SELECT ?org ?orgLabel ?image
WHERE
{
  ?org wdt:P31 wd:Q4830453. #instance of orgs
  ?org wdt:P18 ?image #has image
  
  SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot;}
}&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23List%20of%20organisations%20%0A%0ASELECT%20%3Forg%20%3ForgLabel%20%3Fimage%0AWHERE%0A%7B%0A%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%20%23instance%20of%20orgs%0A%20%20%3Forg%20wdt%3AP18%20%3Fimage%0A%20%20%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%7D%0A%7D SPARQL-query], 2913 Results.

It can be concluded that the number of organizations with the image is 2913. This is not so much, which indicates about incompleteness of information.

Let's build a table of (maybe) popular user requests for organizations (depending on who is interested in some things about the organization). Also, we sort it by descending the results.

&lt;table border=&quot;1&quot;&gt;
   &lt;caption&gt;Table of requests in Wikidata&lt;/caption&gt;
   &lt;tr&gt;
    &lt;th&gt;Request name&lt;/th&gt;
    &lt;th&gt;Quantity of results&lt;/th&gt;
   &lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;inception&lt;/td&gt;&lt;td&gt;30995&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;founded by&lt;/td&gt;&lt;td&gt;5722&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;subsidiary&lt;/td&gt;&lt;td&gt;3398&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;subsidiary&lt;/td&gt;&lt;td&gt;2913&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;location&lt;/td&gt;&lt;td&gt;577&lt;/td&gt;&lt;/tr&gt;
   &lt;tr&gt;&lt;td&gt;motto&lt;/td&gt;&lt;td&gt;2&lt;/td&gt;&lt;/tr&gt;
  &lt;/table&gt;

The results of this table indicate that the quantity of necessary information about organizations is very small, considering their total number on the Wikidata.

There is an opportunity to investigate organizations in Russia too.
We can try to get a list of organizations in Russia with the help of the Wikidata.

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#List of organizations 

SELECT ?org ?orgLabel
WHERE
{
  ?org wdt:P31 wd:Q4830453. #instance of organizations
  ?org wdt:P17 wd:Q159. #Russia country

  SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot;}
}&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23List%20of%20organisations%20%0A%0ASELECT%20%3Forg%20%3ForgLabel%20%3Flocation%0AWHERE%0A%7B%0A%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%20%23instance%20of%20orgs%0A%20%20%3Forg%20wdt%3AP17%20wd%3AQ159.%20%23Russia%20country%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%7D%0A%7D SPARQL-query], 577 Results.

There are 577 organizations that were output by the query. For example, the user wants to see how these organizations are located on the map. It is necessary to write a script.

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#Map of organizations 
#defaultView:Map

SELECT ?org ?orgLabel ?location
WHERE
{
  ?org wdt:P31 wd:Q4830453. #instance of orgs
  ?org wdt:P17 wd:Q159. #Russia country
  ?org wdt:P625 ?location #display location

  SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot;}
}&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23List%20of%20organisations%20%0A%23defaultView%3AMap%0A%0ASELECT%20%3Forg%20%3ForgLabel%20%3Flocation%0AWHERE%0A%7B%0A%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%20%23instance%20of%20orgs%0A%20%20%3Forg%20wdt%3AP17%20wd%3AQ159.%20%23Russia%20country%0A%20%20%3Forg%20wdt%3AP625%20%3Flocation%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%7D%0A%7D SPARQL-query], 9 Results.

Result: very few records with geographic coordinates in Russia. We can get a map of organizations not only in Russia, but of all organizations in the world by using the following script.

&lt;syntaxhighlight lang=&quot;SPARQL&quot;&gt;
#List of organizations 
#defaultView:Map

SELECT ?org ?orgLabel ?location
WHERE
{
  ?org wdt:P31 wd:Q4830453. #instance of orgs
  ?org wdt:P625 ?location

  SERVICE wikibase:label { bd:serviceParam wikibase:language &quot;en&quot;}
}&lt;/syntaxhighlight&gt;

[https://query.wikidata.org/#%23List%20of%20organisations%20%0A%23defaultView%3AMap%0A%0ASELECT%20%3Forg%20%3ForgLabel%20%3Flocation%0AWHERE%0A%7B%0A%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%20%23instance%20of%20orgs%0A%20%20%3Forg%20wdt%3AP625%20%3Flocation%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%7D%0A%7D SPARQL-query], 511 Results.

The result (Fig. 3), again, is very small, only 511 organizations. The quantity of organizations with location is even less than the total number of all organizations in Russia.

[[File:World organizations map.jpg|thumb|Fig. 3: World organizations map|center|900px]]&lt;br style=&quot;clear: both;&quot;&gt;

Analyzing the data obtained, it can be concluded that the information about organizations on the Wikidata are only partially filled. There is not enough information to do any definite conclusions about the organizations and their components. A small amount of information can be explained by the chaotic appearance and disappearance of organizations (it is not easy to survive in such conditions of competition and the existing economy). But the information even about such major organizations (Apple, Microsoft, Intel) is incomplete and needs to be improved (for example, the Intel organization does not have a motto on Wikidata).

== Future work ==
# Output 20 organizations with the largest revenue.
# Output as a diagram how many commercial organizations are appear each year.
# What is the distribution of the quantity of commercial organizations by industry in different countries.

== Test ==
&lt;quiz display=simple&gt;

{ The following commercial organizations are listed: [[w:Tele2|Tele2]], [[w:Lada|Lada]], [[w:Aviakor|Aviakor]], [[w:Uralmash|Uralmash]].
Correlate the organization's data with the images below.
|type=&quot;()&quot;}
|1 (Tele2),|2 (Lada),|3 (Aviakor),|4 (Uralmash)
+--- [[Image:Ростелеком Румянцево.jpg|240px|]] 
---+ [[Image:MainBildingUralmash.jpg|240px|]] 
-+-- [[Image:Lada Kalina (1118).jpg|240px|]] 
--+- [[Image:Tu154-aviakor.jpg|240px|]] 


{ Such commercial organizations are known: MegaFon, [[w:Svyaznoy|Svyaznoy]], [[w:EurosetEvroset]], Sportmaster. Years of the creation of commercial organizations are known: 1992, 1995, 1997, 2002. &lt;br&gt;
Arrange the organization's data in order of increasing date of their creation (1st place is the oldest organization, 4th place is the newest one).&lt;br&gt;
|type=&quot;()&quot;}
|1 place (1992),|2 place (1995),|3 place (1997),|4 place (2002)
---+ [[Image:MegaFon logo Russian.svg|120px|]] MegaFon
-+-- [[Image:SvyaznoyLogo.png|120px|]] Svyaznoy
--+- [[Image:Euroset.png|120px|]] Evroset
+---Sportmaster

{ Arrange countries in ascending order of the number of organizations (on the 1st place: least number of organizations):
|type=&quot;()&quot;}
| 1 | 2 | 3 | 4
-+-- Sweden 
+--- United Kingdom
---+ USA
--+- Germany

&lt;/quiz&gt;

SPARQL-queries with answers:
*
[https://query.wikidata.org/#%23List%20of%20organizations%20%0ASELECT%20%3Forg%20%3ForgLabel%20%0AWHERE%0A%7B%0A%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%20%23instance%20of%20organizations%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%7D%0A%7D List of all organizations],
*
[https://query.wikidata.org/#%23List%20of%20organisations%20%0A%0ASELECT%20%3Forg%20%3ForgLabel%20%3Finception%0AWHERE%0A%7B%0A%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%20%23instance%20of%20orgs%0A%20%20%3Forg%20wdt%3AP571%20%3Finception%0A%20%20%20%20%20%20%20%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%7D%0A%7D List of all organizations with years of creation],
*
[https://query.wikidata.org/#%23List%20of%20organisations%20%0A%0ASELECT%20%3Forg%20%3ForgLabel%20%3Fimage%0AWHERE%0A%7B%0A%20%20%3Forg%20wdt%3AP31%20wd%3AQ4830453.%20%23instance%20of%20orgs%0A%20%20%3Forg%20wdt%3AP17%20wd%3AQ159.%20%23country%20%3D%20Russia%0A%20%20%3Forg%20wdt%3AP18%20%3Fimage%0A%20%20%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22%7D%0A%7D List of all organizations In Russia with image],
*
[https://query.wikidata.org/#SELECT%20%3Forg%20%3Fcountry%20%28count%28%2a%29%20as%20%3Fcount%29%0AWHERE%0A%7B%0A%09%3ForgLabel%20wdt%3AP31%20wd%3AQ4830453%3B%0A%20%20%20%20%20%20%20%20wdt%3AP17%20%3Forg.%0A%20%20%09OPTIONAL%20%7B%0A%09%09%3Forg%20rdfs%3Alabel%20%3Fcountry%0A%09%09filter%20%28lang%28%3Fcountry%20%29%20%3D%20%22ru%22%29%0A%09%7D%0A%7D%0AGROUP%20BY%20%3Forg%20%3Fcountry%0AORDER%20BY%20DESC%28%3Fcount%29%20%0A List of organizations by country in descending order]

== References ==

*{{cite web
|url=https://www.nalog.ru/rn77/service/egrip2/
|title = Access to EGRUL and EGRIP
|year          = 2017
|ref          = {{harvid|EGRUL|2017}}
}}

*{{cite web
|last1       = Andrew Krizhanovsky, Nikita Nikolaev
|title    = Коммерческие организации
|trans-title = Business Enterprise
|url       = https://www.authorea.com/users/86022/articles/177807-wd-business-enterprise2
|publisher     = Authorea
|year          = 2017
}}

[[Category:Research in programming Wikidata|{{SUBPAGENAME}}]]
[[Category:Business]]</text>
      <sha1>58rgm6ixydo8y6w9hrf0xy49x1xtoyr</sha1>
    </revision>
  </page>
  <page>
    <title>User:Jtneill/Wikimedia</title>
    <ns>2</ns>
    <id>233950</id>
    <revision>
      <id>2816388</id>
      <parentid>2811244</parentid>
      <timestamp>2026-06-21T10:28:18Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ Artificial intelligence</comment>
      <origin>2816388</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="688" sha1="n8nv14a9lem272ix64gmcajxoapif6i" xml:space="preserve">==Art and museums==
* [[outreach:GLAM/Newsletter|GLAM]] (Galleries Libraries Archives Museums) | [[Outreach:GLAM/Newsletter/Archives|Archives]]

==Artificial intelligence==
* [[meta:Artificial intelligence]]
** [[meta:Artificial intelligence/2026 Wiki AI]]

==Education==
* [[Wikimedia Education]]
** [[outreach:Special:PermanentLink/45658|Ideas about Wikimedia Education]] (2013)
* [[meta:Learning_and_Evaluation/Newsletter|Learning Quarterly]]
* [[outreach:Education/Newsletters|This Month in Education]] | [[Outreach:Education/Newsletter/Archives|Archives]]

==Research==
* [[meta:Research:Newsletter]]

==Technology==
* [[Wikiversity:Newsletters/Tech News]]

[[Category:User:Jtneill]]</text>
      <sha1>n8nv14a9lem272ix64gmcajxoapif6i</sha1>
    </revision>
  </page>
  <page>
    <title>Universal Bibliography/Countries</title>
    <ns>0</ns>
    <id>269370</id>
    <revision>
      <id>2816371</id>
      <parentid>2806334</parentid>
      <timestamp>2026-06-21T02:34:28Z</timestamp>
      <contributor>
        <username>James500</username>
        <id>297601</id>
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      <text bytes="35900" sha1="79j3l7cjqaa53qhm5wsm5vppkrwkhk0" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Geography|Geography]].

See [[w:Category:Bibliographies of countries or regions]] and [[w:Category:Works about countries]].

This part of the [[Universal Bibliography]] is a bibliography of countries (including former countries).

==Countries==
*Bateman and Egan (eds). The Encyclopedia of World Geography: A Country by Country Guide. 1993. Revised 1997.
*Peter Stalker. Handbook of the World. 2000. A Guide to Countries of the World. (Oxford Guide to Countries of the World. 2nd Ed: 2004, 2nd Revised Ed: 2007 [https://books.google.co.uk/books?id=GtztAAAAMAAJ], 3rd Ed: 2010 [https://books.google.co.uk/books?id=gvKvfxkbZ1AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Countries of the World and Their Leaders Yearbook. Gale. [https://books.google.co.uk/books?id=5etKAAAAYAAJ] [https://books.google.co.uk/books?id=p41OAAAAIAAJ]
*Hutchinson Guide to Countries of the World [https://books.google.co.uk/books?id=GgpjUe4kN_IC]
*The World Guide: Global Reference, Country by Country. 11th Ed: 2007 [https://books.google.co.uk/books?id=EoWoLgAACAAJ]
*Spence. The World Today: A Nation-by-Nation Guide. Cassell. 1994. 1999. [https://books.google.com/books?id=Ub8qOQAACAAJ]
*Worldmark Encyclopedia of the Nations [https://books.google.co.uk/books?id=I0oYAQAAMAAJ]
*Kurian. Encyclopedia of the World's Nations. Facts on File. Reviews: [https://books.google.co.uk/books?id=Y1EnAQAAIAAJ] [https://books.google.co.uk/books?id=lz0RAQAAMAAJ]
*Michael O'Mara. Facts about the World's Nations. 1999. [https://books.google.co.uk/books?id=mygYAAAAIAAJ]
*Status of the World's Nations. 1965 [https://books.google.co.uk/books?id=sftEyRbAXMUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 1973 [https://books.google.co.uk/books?id=kw2U_Cg2gKYC&amp;pg=PP3#v=onepage&amp;q&amp;f=false].
*[[s:Author:John Alexander Hammerton|Hammerton, John Alexander]] (ed). Countries of the World. Published at the Fleetway House. 6 vols. [https://books.google.co.uk/books?id=e6IaAQAAMAAJ] [https://books.google.co.uk/books?id=K5oaAQAAMAAJ]
*[[s:Author:Robert Brown (1842-1895)|Brown, Robert]]. The Countries of the World. [https://books.google.co.uk/books?id=nO0DAAAAQAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*A Morely Dell. The Countries of the World. (Harrap's New Geographical Series). 1932. (School certificate). Reviews: [https://books.google.co.uk/books?id=oSS9PB_Jf7AC] [https://books.google.co.uk/books?id=BicVAAAAIAAJ] [https://books.google.co.uk/books?id=5qBOAAAAIAAJ] [https://books.google.co.uk/books?id=YbwcAQAAIAAJ] [https://books.google.co.uk/books?id=sc1AAAAAIAAJ]

General series:
*National Geographic Countries of the World [https://books.google.co.uk/books?id=IT2wfzVIPykC]
*Countries of the World. Evans Brothers. (GCSE) [https://books.google.co.uk/books?id=a3sZvWc7E1EC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*One Europe. Longman. [https://search.worldcat.org/en/title/west-germany-adapted-by-lj-russon-from-the-original-german-by-sylvia-lof-ingrid-mallberg-dietrich-rosenthal/oclc/561591761]
*Collier's Nations of the World. The Nations of the World: An Historical Series. [https://books.google.co.uk/books?id=VJY-AAAAYAAJ&amp;pg=PP8#v=onepage&amp;q&amp;f=false]
*Collier's History of Nations. The History of Nations. [https://books.google.co.uk/books?id=fmSUfTY5E80C]
*The Story of the Nations. T Fisher Unwin.
*The World and Its Peoples. (The Illustrated Library of the World and Its Peoples). Greystone Press, New York.
*World and Its Peoples. Marshall Cavendish. [https://books.google.co.uk/books?id=oms5xjI7ba0C&amp;pg=PA141#v=onepage&amp;q&amp;f=false]

==England==
===Counties===
See [[s:Portal:Counties]]

* Harrison, &quot;County Bibliography&quot; (1886) 3 Library Chronicle [https://books.google.co.uk/books?id=Wz9FAAAAYAAJ&amp;pg=PA49#v=onepage&amp;q&amp;f=false 49]

General series
*Victoria County History
*Oxford County Histories
*Pinnock's County Histories
*Shire County Guides. Shire Publications.
*Cambridge County Geographies
*Pike's New Century Series
*[[s:Page:County Churches of Cornwall.djvu/6|County Churches]]. G Allen.

Avon
*Moore. Avon Local History Handbook. Phillimore. 1979. [https://books.google.co.uk/books?id=h0kjAAAAMAAJ] Bibliography, p 102

Bedfordshire
*Conisbee, Lewis Ralph. A Bedfordshire Bibliography. Bedfordshire Historical Record Society. Bedford. 1962. Supplements 1967, 1971, 1978. Third supplement by Threadgill. Review: 6 Archives 52 [https://books.google.co.uk/books?id=oOMZAAAAYAAJ]. See also [https://books.google.co.uk/books?id=MjspAAAAYAAJ] [https://books.google.co.uk/books?id=PejgAAAAMAAJ]
*Godber. History of Bedfordshire. 1969. 1984. [https://books.google.co.uk/books?id=jdvwPQAACAAJ]
*Pinnock. The History and Topography of Bedfordshire [https://books.google.co.uk/books?id=9bJYAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Parry. Select Illustrations, Historical and Topographical, of Bedfordshire [https://books.google.co.uk/books?id=UTUJAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Blyth. The History of Bedford and Visitor's Guide. 1873 [https://books.google.co.uk/books?id=IuIGAAAAQAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Cambridge County Geographies [https://books.google.co.uk/books?id=kTc8AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Buckinghamshire
*Reed. A History of Buckinghamshire. 1993 [https://books.google.co.uk/books?id=BtkWAQAAIAAJ]

Cambridgeshire
*Carter. History of the County of Cambridge [https://books.google.co.uk/books?id=jXpbAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Babington. Ancient Cambridgeshire [https://books.google.co.uk/books?id=DPrCAwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Devon
*Ravenhill and Rowe. Devon Maps and Map-makers [https://books.google.co.uk/books?id=tjf2yAEACAAJ]
*Wright. A Plea for a Devonshire Bibliography. 1885 [https://books.google.co.uk/books?id=8ZUDAAAAQAAJ]

Derbyshire
*Woore. A Catalogue of Local Maps of Derbyshire, C.1528-1800. 2012. [https://books.google.co.uk/books?id=oWmCMwEACAAJ]
*O'Neal. A Bibliography of Derbyshire Lead Mining. 1961

Essex
*Cunnington. Catalogue of Books, Maps and Manuscripts, relating to or connected with the County of Essex. 1902 [https://books.google.co.uk/books?id=oIcqpibGE4MC]
*&quot;The Bibliography of Essex&quot; (1882) 1 Antiquarian Magazine &amp; Bibliographer [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false 72]. See also [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA283#v=onepage&amp;q&amp;f=false p 283].
*&quot;The Bibliography of Essex&quot; (1891) 5 The Essex Naturalist 30 [https://books.google.co.uk/books?id=iIo1AQAAMAAJ]
*Moon. Essex Literature. 1900. Review: 61 Literary World 438 [https://books.google.co.uk/books?id=2T0ZAAAAYAAJ] See also [https://books.google.co.uk/books?id=1Y4UAQAAIAAJ] [https://books.google.co.uk/books?id=C_pEAAAAMAAJ]
*Fenn and Lowery, &quot;An Essex Bibliography&quot;, Journal of the South West Essex Technical College, vols 2 &amp; 3
*Victoria County History bibliography. 1959 [https://books.google.co.uk/books?id=F2EJAQAAIAAJ]
*O'Leary, John Gerard. A Supplement to the Essex Bibliography. Dagenham. 1962.
*A Bibliography of Essex Archaeology &amp; History
*Essex and Dagenham: A Catalogue of Books, Pamphlets and Maps. Dagenham. 1961
*Essex Archaeology and History: The Transactions of the Essex Society for Archaeological and History [https://books.google.co.uk/books?id=CtFAAAAAYAAJ]
*Essex Naturalist: Being the Journal of the Essex Field Club
*Wright. The History and Topography of the County of Essex [https://books.google.co.uk/books?id=SgQVAAAAQAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Ogborne, The History of Essex [https://books.google.co.uk/books?id=IeVSAAAAcAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Suckling. Memorials of the Antiquities and Architecture, Family History and Heraldry of the County of Essex [https://books.google.co.uk/books?id=bcw_AAAAcAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Hunter, The Essex Landscape: A Study of Its Form and History [https://books.google.co.uk/books?id=w9kWAQAAIAAJ]
*Cambridge County Geography [https://books.google.co.uk/books?id=GPHa_X_0qo0C&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Sokoll. Essex  Pauper Letters, 1731-1837 [https://books.google.co.uk/books?id=rCLia7XlqtMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Morant. The History and Antiquities of Colchester in the County of Essex [https://books.google.co.uk/books?id=DDgtAAAAYAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Wallen. The History and Antiquities of the Round Church at Little Maplestead, Essex [https://books.google.co.uk/books?id=FPYVAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Kent
*Smith. Bibliotheca Cantiana. 1837. [https://books.google.co.uk/books?id=1dJDAAAAYAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false]

Leicestershire
*Kirkby, C V (compiler). Catalogue of the books, pamphlets, &amp;c., relating to Leicestershire in the Central Reference Library. Leicester Free Public Libraries. 1893. Reviews: [https://books.google.co.uk/books?id=3boqAQAAIAAJ&amp;pg=PA84#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=UcHnAAAAMAAJ&amp;pg=PA728#v=onepage&amp;q&amp;f=false]
*Leicestershire and Rutland Bibliography, 1963-65 (1966) [https://books.google.co.uk/books?id=-OhVAAAAYAAJ 40] Leicestershire Archaeological and Historical Society: Transactions (1964/5) 92. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1961-63. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1960-61. Available as pdf from University of Leicester.
*A Bibliography of the Small Towns in Leicestershire and Rutland, 1600–1850. (Dissertation). [https://repository.lboro.ac.uk/articles/educational_resource/A_bibliography_of_the_small_towns_in_Leicestershire_and_Rutland_1600_1850/9414200]
*Loughborough's Heritage: A Bibliography of the Holdings of Leicestershire Libraries and Information Service and Record Office. [https://books.google.co.uk/books?id=Bwx2zgEACAAJ]
*Keith Ambrose and Frank Williams, &quot;Bibliography of the Geology of Leicestershire and Rutland: Part 2: 1971-2003&quot; (2004) [https://books.google.co.uk/books?id=U-tQAQAAIAAJ 16] The Mercian Geologist 5. Available as pdf from East Midlands Geological Society.
*Parsons and Brandwood. A Bibliography of Leicestershire Churches. 1978.
*Education in Leicestershire: A Bibliography. [https://books.google.co.uk/books?id=X6EfzQEACAAJ]

Sussex
*Brent, Fletcher and McCann. Sussex in the 16th and 17th Centuries: A Bibliography. 2nd Ed [https://books.google.co.uk/books?id=I7UtAAAAYAAJ]
*Farrant. Sussex in the 18th and 19th Centuries: A Bibliography. 1st Ed: 1973, 2nd Ed: 1977 [https://books.google.co.uk/books?id=MLUtAAAAYAAJ], 3rd Ed: 1979

==France==
Bibliography:
*Bibliographie de la France. Commentary: Encyclopedia of Library and Information Science, vol 37, supplement 2, [https://books.google.co.uk/books?id=10rgjNvOV8oC&amp;pg=PA145#v=onepage&amp;q&amp;f=false p 145]; The Bookseller, 6 January 1881, [https://books.google.co.uk/books?id=4dsiAQAAMAAJ&amp;pg=PA10#v=onepage&amp;q&amp;f=false p 10]; Stein, Manuel de bibliographie générale, [https://books.google.co.uk/books?id=lJYPyKjV1qYC&amp;pg=PA23#v=onepage&amp;q&amp;f=false p 23]. 
*Girault de Saint-Fargeau. Bibliographie historique et topographique de la France. 1845 [https://books.google.co.uk/books?id=kClB9CQNZoMC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Catalogue d'une collection d'ouvrages sur l'histoire des provinces de la France. 1842 [https://books.google.co.uk/books?id=qQBX5WZouzAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Landscape:
*Beaujeu-Garnier. France. (The World's Landscapes). 1975. [https://books.google.com/books?id=nwxDAQAAIAAJ]

Agenais:
*Andrieu. Bibliographie générale de l’Agenais et des parties du Condomois et du Bazadais. 1886 to 1891. Reprinted 1969.

Alsace:
*Ristelhuber. Bibliographie alsacienne. 1869 to 1873 [https://books.google.co.uk/books?id=0mhLAQAAMAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Bibliographie alsacienne: Revue critique des publications concernant l'Alsace. 1918 to 1936
*Ritter. Répertoire bibliographique des livres imprimés en Alsace aux XVe et XVIe siècles [https://books.google.co.uk/books?id=DewaAQAAMAAJ]

Angoumois:
*Castaigne. Essai d'une bibliothèque historique de l'Angoumois, ou Catalogue raisonné des principaux ouvrages qui traitent des différentes branches de l'histoire de cette province. 1847 [https://books.google.co.uk/books?id=R-UanmmlvAEC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Anjou:
*Braguier and Braguier. Archéologie en Anjou: bibliographie. 1984 [https://books.google.co.uk/books?id=LvsmAQAAIAAJ]

Auvergne:
*Gonot. Catalogue des ouvrages imprimés et manuscrits concernant l'Auvergne, extrait du catalogue général de la Bibliotlèque de Clermont-Fd (Puy-de-Dome). 1849. [https://books.google.co.uk/books?id=yCFtbObRCbUC&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Catalogue des livres et estampes concernant l'ancienne Province d'Auvergne (Puy-de-Dôme, Cantal, Haute-Loire) réunis par feu M. G. Desbouis. 1865. [https://books.google.co.uk/books?id=Ui4S8_D0N74C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Béarn
*&quot;Bibliographie Béarnaise&quot;, Revue de Pau et du Béarn [https://books.google.co.uk/books?id=FuZnAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=FQYqvPo9D9IC&amp;pg=PA158#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RL9VAAAAYAAJ]

Brittany
*Sacher. Bibliographie de la Bretagne, ou Catalogue général des ouvrages historiques, littéraires et scientifiques parus sur la Bretagne, avec la liste des revues publiées en cette province, les prix approximatifs des volumes rares, etc. 1881 [https://archive.org/details/bibliographiede00sach]

Burgundy:
*Milsand. Bibliographie bourguignonne; ou, Catalogue méthodique d'ouvrages relatifs à la Bourgogne: Sciences - Arts - Histoire. 1885 [https://archive.org/details/bibliographiebo00milsgoog] [https://archive.org/details/bibliographiebo00sciegoog] [https://books.google.co.uk/books?id=CxIIAAAAQAAJ]
*Catalogue des manuscrits de la Bibliothèque royale des ducs de Bourgogne. 1842 [https://books.google.co.uk/books?id=FX5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*The Companion Guide to Burgundy [https://books.google.co.uk/books?id=NraRP0AkDT0C&amp;pg=PP3#v=onepage&amp;q&amp;f=false]
*Lecat. The Golden Book of Burgundy. (The Golden Book) [https://books.google.co.uk/books?id=FyzR9qU1Zl4C&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Gwynn. Burgundy: With Chapters on the Jura and Savoy. (Kitbag Travel Books). 1935 [https://books.google.co.uk/books?id=ny1LAAAAMAAJ]
*Bazin. Wonderful Burgundy. 1988. 1997 [https://books.google.co.uk/books?id=Yt1CRdICWCUC]
*Bailey. Burgundy. (Insight Guides). 1993 [https://books.google.co.uk/books?id=Q69a1dMW2NQC]
*Dunlop. Burgundy. Hamilton.1990 [https://books.google.co.uk/books?id=S_1OAAAAMAAJ]

Champagne:
*Lhermitte. Ouvrages sur la Champagne: contribution à la bibliographie champenoise. 1992. [https://books.google.co.uk/books?id=jbPfAAAAMAAJ]

Dauphiné:
*Mélanges biographiques et bibliographiques relatifs à l'histoire littéraire du Dauphiné par Colomb de Batines et Ollivier Jules. 1837 [https://books.google.co.uk/books?id=2F5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Lorraine:
*Bibliographie lorraine. Académie nationale de Metz [https://books.google.co.uk/books?id=n-DfAAAAMAAJ]

Maine:
*Desportes. Bibliographie du Maine, précédée de la description topographique et hydrographique du diocése du Mans, Sarthe et Mayenne. 1844. [https://books.google.co.uk/books?id=hSk-AAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Normandy:
*Frère. Manuel du bibliographe Normand ou dictionnaire bibliographique et historique. 1858 to 1860. [https://books.google.co.uk/books?id=dp6geJClg1YC&amp;pg=PP13#v=onepage&amp;q&amp;f=false vol 1]

==Japan==
Bibliography
*Jozef Rogala. A Collector's Guide to Books on Japan in English: An Annotated List of Over 2500 Titles with Subject Index. 2001. [https://books.google.co.uk/books?id=7KI9ao-w2FEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ria Koopmans-de Bruijn. Area Bibliography of Japan. (Scarecrow Area Bibliographies). Scarecrow Press. 1998. [https://books.google.co.uk/books?id=Hlx2OMjgUi0C&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Frank Joseph Shulman. Japan. (World Bibliographical Series, vol 103). Clio Press. 1989. [https://books.google.co.uk/books?id=LsoUAQAAIAAJ]
*Eibun Nihon Kankei Tosho Mokuroku, 1945-1981. (Japanese: 英文日本関係図書目録, 1945-1981). (English: Catalogue of Books in English on Japan, 1945-1981). Japan Foundation. Tokyo. 1986.
*Japan: analytical bibliography: with supplementary research aids: and selected data on Okinawa . . . Department of the Army. Washington. 1972. [https://books.google.co.uk/books?id=h4d4nYxrxtMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Books on Japan in Western Languages. The International Christian University Library. 1971. [https://books.google.co.uk/books?id=F2bQAAAAMAAJ]
*Books on Japan: A List of Acquisitions, 1955-1970. International House of Japan Library. 1971. [https://books.google.co.uk/books?id=F8sWAQAAIAAJ]
*Fukuda. Union Catalog of Books on Japan in Western Languages. 1968. [https://books.google.co.uk/books?id=HKYyAQAAIAAJ]
*A Classified List of Books in Western Languages Relating to Japan. University of Tokyo Press. 1965. [https://books.google.co.uk/books?id=U8MUAQAAIAAJ]
*Katsuji Yabuki (ed). Japan Bibliographic Annual. Published by the Hokuseido Press for the Japan Writers Society. 1956 and 1957.
**Japan Bibliographic Annual 1956. [https://books.google.co.uk/books?id=9XLQAAAAMAAJ]
**Japan Bibliographic Annual 1957. [https://books.google.co.uk/books?id=vesSAAAAIAAJ]. Reviews: (1957) 13 Monumenta Nipponica 166 (April-July) [https://books.google.co.uk/books?id=8S1yb-iwrOwC] (1957) 25 The Oriental Economist 212 (April) [https://books.google.co.uk/books?id=QELoAAAAMAAJ]
*Haring. Books on Japan: A Reference List. 1955. [https://books.google.co.uk/books?id=RbDoAAAAMAAJ]
*Borton. A Selected List of Books and Articles on Japan in English, French, and German. 1940: [https://books.google.co.uk/books?id=YYIsAAAAYAAJ]. Revised and enlarged. Harvard University Press. 1954: [https://books.google.co.uk/books?id=F8O2VwJUPUkC].
**A Selected List of Books on Japan in Western Languages (1945-1960). (Studies on Asia Abroad, vol 1). The Information Centre of Asian Studies, The Toyo Bunko. 1964. [https://books.google.co.uk/books?id=i1_QAAAAMAAJ]
*Oskar Nachod. Bibliography of the Japanese Empire 1906-1926. 1928. [https://archive.org/details/bibliographyofja0001oska/page/n8/mode/1up vol 1]. [https://archive.org/details/bibliographyofja0002oska/page/n6/mode/1up vol 2].
*Fr. von Wenckstern. A Bibliography of the Japanese Empire: being a Classified List of All Books, Essays and Maps in European Languages relating to Dai Nihon (Great Japan) published in Europe, America and in the East from 1859-93 . . . 1895. vol 1. [https://books.google.co.uk/books?id=dcVAAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=v7lO4ddqDywC&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**Volume 2, from 1894 to the middle of 1906. 1907. [https://archive.org/details/bibliographyofja0002frvo/page/n6/mode/1up]
*Hyman Kublin. What Shall I Read on Japan? An Introductory Guide. Japan Society, New York. 1971. [https://books.google.co.uk/books?id=yRRUAAAAYAAJ]

Japanese studies
*An Introductory Bibliography for Japanese Studies. The Japan Foundation. [https://books.google.co.uk/books?id=53O6AAAAIAAJ]
*Richard Perren. Japanese Studies from Pre-History to 1990: A Bibliographical Guide. 1992. [https://books.google.co.uk/books?id=CN9RAQAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies&quot; at pp 1 to 3.
*K.B.S. Bibliography of Standard Reference Books for Japanese Studies, with Descriptive Notes. University of Tokyo Press. [https://books.google.co.uk/books?id=95wbAAAAMAAJ]
*[[w:en:Japan Forum]]. British Association for Japanese Studies. [https://www.tandfonline.com/journals/rjfo20]

History and culture
*John W Dower. Japanese History &amp; Culture from Ancient to Modern Times: Seven Basic Bibliographies. 1986. [https://books.google.co.uk/books?id=NX67AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies &amp; Research Guides&quot; at chapter 6.

Research guides
*Mindy L Kotler. Information Gathering on Japan: A Primer. Search Associates. 1988. ISBN 9780962546006. Catalogue: [https://search.worldcat.org/zh-cn/title/Information-gathering-on-Japan-Joho-:-a-primer/oclc/20530148]. Review: (1989) [https://books.google.co.uk/books?id=NZLiAAAAMAAJ 27] Choice 82

Encyclopedias

See also [[w:ja:Japanese encyclopedias]]
*Louis-Frédéric. Japan Encyclopedia. 2002. [https://books.google.co.uk/books?id=p2QnPijAEmEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Japan: An Illustrated Encyclopedia. Kodansha. 1993.
**Japan: Profile of a Nation. Kodansha. 1995. Revised Edition. 1999.
*[[w:Kodansha Encyclopedia of Japan|Kodansha Encyclopedia of Japan]]. 1983. Supplement. 1986. [https://books.google.co.uk/books?id=WvApAQAAMAAJ]
*Dorothy Perkins. Encyclopedia of Japan: Japanese History and Culture, from Abacus to Zori. Facts on File. A Roundtable Press Book. 1991. [https://books.google.co.uk/books?id=JLKGAAAAIAAJ]
*Pictorial Encyclopedia of Modern Japan. Gakken. 1986. [https://books.google.co.uk/books?id=0FgKAQAAIAAJ]
*Boye Layfayette De Mente. Japan Encyclopedia. 1995. [https://books.google.co.uk/books?id=f9c7AAAAMAAJ]
**Boye De Mente. Everything Japanese. [The Authoritave Reference on Japan Today]. 1989. [https://books.google.co.uk/books?id=Duku89bARgoC]

Media
*[https://www.bbc.com/news/world-asia-pacific-15217593 Japan media guide]. News. BBC. 20 March 2023.
*Masaaki Kasagi. Mass Media in Japan. (Orientation seminars on Japan, number 14). 1983. [https://books.google.co.uk/books?id=odkgAAAAIAAJ]
*Routledge Handbook of Japanese Media [https://books.google.co.uk/books?id=zilKDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Publishers
*[https://www.publishersweekly.com/pw/by-topic/international/international-book-news/article/99729-get-to-know-these-japanese-publishing-companies.html Get to Know These Japanese Publishing Companies]. Publishers Weekly. 20 February 2026.

Press and journalism
*[https://reutersinstitute.politics.ox.ac.uk/digital-news-report/2025/japan Japan]. Reuters Institute for the Study of Journalism. 17 June 2025.
*Marjane Aalam and Philippe Régnier. The Japanese Press and Information System. The Graduate Institute of International Studies. Geneva. [https://books.google.co.uk/books?id=RTcbAQAAIAAJ]
*The Japanese Press: Past and Present. Japan Newspaper Publishers' and Editors' Association. [https://books.google.co.uk/books?id=5tcQAAAAIAAJ 1949].
*Anthony Rausch. Japanese Journalism and the Japanese Newspaper: A Supplemental Reader. [https://books.google.co.uk/books?id=mZrToQEACAAJ]
*Frank L Martin. The Journalism of Japan. 1918. [https://books.google.com/books?id=ruYzAQAAMAAJ]
*William De Lange. A History of Japanese Journalism. Japan Library. 1998. [https://books.google.co.uk/books?id=Rd5tb0cuz8QC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kanesada Hanazono. The Development of Japanese Journalism. Osaka. 1924. [https://books.google.co.uk/books?id=z99ZAAAAMAAJ]
*Kanesada Hanazono. Journalism in Japan and Its Early Pioneers. 1926. [https://books.google.co.uk/books?id=IGTFfLc4bq0C]
*César Castellvi. A Sociology of Journalism in Japan: The Last Empire of the Press. 2024. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PR4#v=onepage&amp;q&amp;f=false]
*&quot;Japan&quot;. Christopher H Sterling (ed). Encyclopedia of Journalism. A Sage Reference Publication. 2009. ISBN 9780761929574. vol 3. pp [https://books.google.co.uk/books?id=ZQhDq8fPj2IC&amp;pg=PA809#v=onepage&amp;q&amp;f=false 809] to 815.

Press annuals
*The Japanese Press. (Nihon Shinbun Kyokai). [https://books.google.co.uk/books?id=AfvyAAAAMAAJ 1979] [https://books.google.co.uk/books?id=Au3yAAAAMAAJ 1998]

Summaries of the press
*Daily Summary of Japanese Press

Foreign correspondents
*Foreign Correspondents in Japan: Reporting a Half Century of Upheavals, from 1945 to the Present. Tuttle. 1998. [https://books.google.co.uk/books?id=YI3TAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Nunn (comp). Japanese Periodicals and Newspapers in Western Languages: An International Union List. Mansell. 1979. [https://books.google.co.uk/books?id=jEROAQAAIAAJ]
*Japan Periodicals. Keizai Koho Center. 3rd Ed [https://books.google.co.uk/books?id=ATm0AAAAIAAJ]. Japan Periodicals, 1982. [https://books.google.co.uk/books?id=PkMyAAAAMAAJ]
*Japanese Periodicals Index
**Humanities and Social Sciences [https://books.google.co.uk/books?id=nXX_RpPGf3AC]
**Natural Sciences [https://books.google.co.uk/books?id=FCJIAAAAYAAJ]
*Current Japanese Periodicals [https://books.google.co.uk/books?id=FjO5AAAAIAAJ]
*Check-list of Japanese Periodicals Held in British University and Research Libraries. [https://books.google.co.uk/books?id=VZgsAAAAYAAJ]
*Union List of Current Japanese Periodicals in the East Asian Libraries of Columbia, Harvard, Princeton, and Yale Universities. [https://books.google.co.uk/books?id=yw7kAAAAMAAJ]
*List of Japanese Periodicals in the Library of the School of Oriental &amp; African Studies. [https://books.google.co.uk/books?id=RREjAQAAIAAJ]
*Gianni Simone. [https://www.japantimes.co.jp/community/2011/04/26/issues/english-mags-approach-milestone-crossroads/ English mags approach milestone, crossroads]. The Japan Times. 26 April 2011.
*Japan Report (1955 onwards) (Consulate General of Japan, Japan Information Center). Vol 39 published in 1993. [https://books.google.co.uk/books?id=MX4BN_frv4IC&amp;pg=PP7#v=onepage&amp;q&amp;f=false] editions:jYuMSMIQC-AC
**Japan Information
*Japan Now [https://books.google.co.uk/books?id=Nul7DRQaexMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Japan Quarterly. (Asahi Shimbun). 1954 to 2001. [https://books.google.co.uk/books?id=nZMMAQAAMAAJ] [https://books.google.co.uk/books?id=_RwVAAAAMAAJ] 189 issues.
*Japan Illustrated: The Japan Times Quarterly [Pictorial] Magazine (October 1963 to Summer 1977) 15 vols [https://books.google.co.uk/books?id=D7UThOmE8T4C]
*[[w:Japan Spotlight|Japan Spotlight]]. Economy, Culture &amp; History: Japan Spotlight: Bimonthly. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ]
*Focus Japan. (Japan External Trade Organization, JETRO). [https://books.google.co.uk/books?id=2fG2hsEZpRkC]
*The Japan Journal [https://books.google.co.uk/books?id=2V3hAAAAMAAJ] [https://books.google.co.uk/books?id=CJwoAQAAMAAJ]
*The Japan Magazine: A Representative Monthly of Things Japanese [https://books.google.co.uk/books?id=ubGKo-p6O_0C] [https://archive.org/details/jm-1914-v4.9-5.2/mode/1up]
*Transactions and Proceedings of the Japan Society, London [https://books.google.co.uk/books?id=B75nnph5qHgC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
**Bulletin. [Bulletin of the Japan Society, London.] [https://books.google.co.uk/books?id=Pd9KvyhnpjMC]
**The Japan Society of London Bulletin [https://books.google.co.uk/books?id=XxlxAAAAMAAJ]
*About Japan. Japan Society, New York. [https://books.google.co.uk/books?id=Nf5OAQAAIAAJ]
**News Bulletin [https://archive.org/details/bub_gb_QcA3AQAAIAAJ/page/n2/mode/1up]
*[[w:en:Metropolis (free magazine)|Metropolis]] (metropolisjapan.com)
*[[w:en:Tokyo Weekender|Tokyo Weekender]] (トーキョー・ウィークエンダー) [https://www.tokyoweekender.com/japan-life/news-and-opinion/nhk-world-features-the-tokyo-weekender-magazine/]
*The Japan Gazette [https://books.google.co.uk/books?id=WSopAAAAYAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Tokio Times [https://books.google.co.uk/books?id=UDfiFBu0vB4C&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*[[w:en:Look Japan|Look Japan]]. (Look Japan Ltd). [https://books.google.co.uk/books?id=QnO6AAAAIAAJ]. Commentary: Gale Directory of Publications and Broadcast Media [https://books.google.co.uk/books?id=ve4dAQAAMAAJ]
*[[w:en:Japan Echo|Japan Echo]]. 1974 to 2010. [https://books.google.co.uk/books?id=Cmq6AAAAIAAJ] [https://books.google.co.uk/books?id=fpmEPpl-85UC]
*PHP Intersect. (Where Japan Meets Asia and the World). PHP Institute. [https://books.google.co.uk/books?id=i74TAQAAMAAJ]
**Intersect Japan [https://books.google.co.uk/books?id=sL8TAQAAMAAJ]
*Speaking of Japan [https://books.google.co.uk/books?id=U7S0AAAAIAAJ]. [Speeches.]
*The Hansei Zasshi: A Monthly Magazine [https://books.google.co.uk/books?id=dyIsvnYjpwEC&amp;pg=PP6#v=onepage&amp;q&amp;f=false]

Newspapers

See also [[w:List of newspapers in Japan]]
*Tanner. English Language Newspapers in Bakumatsu Japan. 1977. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*[https://www.japantimes.co.jp/news/2009/03/03/reference/newspapers-here-soldiering-on/ Newspapers here soldiering on]. The Japan Times. 3 March 2009.

*[[w:The Japan Times|The Japan Times]]
**The Japan Times: Weekly Edition [https://books.google.co.uk/books?id=KoQ-AQAAMAAJ] [https://books.google.co.uk/books?id=yYQ-AQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Japan Daily Mail
*Japan Weekly Mail
*The Japan Chronicle
**Weekly Edition [https://books.google.co.uk/books?id=vXdRAQAAIAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Japan News. (The Japan News by The Yomiuri Shimbun)
**Yomiuri Japan News (from 1955)
**The Yomiuri (from 1958)
**The Daily Yomiuri (from 1970)
*The Asahi Shimbun: Asia &amp; Japan Watch. [https://www.asahi.com/sp/ajw/]
**Asahi Evening News (from 1954)
***Tokyo Evening News (1952 to 1954) [https://ndlsearch.ndl.go.jp/books/R100000002-I000000145073]
*The Mainichi. [https://mainichi.jp/english/]
**Mainichi Daily News (1922 to 2001) [https://www.nytimes.com/2001/02/27/business/worldbusiness/IHT-tech-briefstop-the-presses.html] [https://ndlsearch.ndl.go.jp/books/R100000002-I000000144910]

Sports newspapers; sports dailies
*Louise do Rosario, &quot;News-stand stars&quot; in &quot;Japan&quot; (1992) [https://books.google.co.uk/books?id=T_GzAAAAIAAJ 155] [[w:en:Far Eastern Economic Review|Far Eastern Economic Review]], 24 to 31 December 1992, p 21
*[[w:ja:岡崎満義|Mitsuyoshi Okazaki]], &quot;Unsportsmanlike Journalism: Japan's sports dailies may be popular, but are they sporting?&quot; in &quot;Sport&quot;, [[w:en:Look Japan|Look Japan]], [https://books.google.co.uk/books?id=lD3tAAAAMAAJ January 1995], p 39

News
*[[w:en:Japan Today|Japan Today]] (ジャパントゥデイ). GPlusMedia. Gakken Holdings.

Annuals and year books 
*This is Japan. Asahi Shimbun. 1954 to 1971. [https://books.google.co.uk/books?id=2X9DAQAAIAAJ]. Commentary: A Victorian Sailor's Grave in the Seto Inland Sea, p 244 [https://books.google.co.uk/books?id=OegkAgAAQBAJ&amp;pg=PA244#v=onepage&amp;q&amp;f=false]
*The Japan Year Book. The Japan Year Book Office. 1905 onwards. [https://archive.org/details/bub_gb_arFPAAAAMAAJ/page/n10/mode/1up 1906]. [https://archive.org/details/in.ernet.dli.2015.553496/page/n27/mode/1up 1915].
*The &quot;Japan Gazette&quot; Japan Year Book. The Japan Gazette. [https://archive.org/details/japan-year-book-1913-1914/page/n15/mode/1up 1913-14]
*The Japan Times Year Book

Almanacs
*Asahi Shimbun Japan Almanac. [https://books.google.co.uk/books?id=SEEEAQAAIAAJ 1995].
*Japan Almanac. (The Mainichi Newspapers). [https://books.google.co.uk/books?id=ufAIAQAAIAAJ 1972]. [https://books.google.co.uk/books?id=X4eXWRkbtFsC 1973]. [https://books.google.co.uk/books?id=7rMrAAAAIAAJ] [https://books.google.co.uk/books?id=krMrAAAAIAAJ]
*[[w:Boyé Lafayette De Mente|Boye De Mente]]. Passport's Japan Almanac. [https://books.google.co.uk/books?id=741wAAAAMAAJ]

General
*Japan: A Country Study. (Area Handbook series). 4th Ed: 1983: [https://books.google.co.uk/books?id=HkM5N3JNc5IC]. 5th Ed: 1992: [https://books.google.co.uk/books?id=ze-wupXxpvEC]
*Area Handbook for Japan. 2nd Ed: 1964: [https://books.google.co.uk/books?id=WucdAAAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]. 3rd Ed: 1974: [https://books.google.co.uk/books?id=LG2aoq1U_eoC&amp;pg=PR1#v=onepage&amp;q&amp;f=false] (DA Pam 550-30).
*Colin Simpson. Picture of Japan.
**Japan: An Intimate View. A S Barnes. [https://books.google.co.uk/books?id=3hkeAAAAMAAJ]
**This is Japan. Angus &amp; Robertson. [https://books.google.co.uk/books?id=HJEJAQAAIAAJ]
*Japan. (The World and Its Peoples). Greystone Press, New York. 1964. Volume 1: [https://books.google.co.uk/books?id=yysUAQAAMAAJ]. Volume 2 &quot;Japan Korea&quot;, including Korea: [https://books.google.co.uk/books?id=uQAUAQAAMAAJ]. See pp 1 to 375 for Japan, and pp 376 to 379 for Ryukyu and Bonin Islands.
*Japan. (World and its Peoples: Eastern and Southern Asia, volume 8). Marshall Cavendish. 2008. ISBN 9780761476412.
*Edward Seidensticker. This Country, Japan. Kodansha International. 1979. ISBN 9780870112294. [https://books.google.co.uk/books?id=88wwAQAAIAAJ]
*Hall and Beardsley. Twelve Doors to Japan. McGraw-Hill. New York. 1965. [https://books.google.co.uk/books?id=0KpxAAAAMAAJ]

Handbooks
*Heenan (ed). The Japan Handbook. (Regional Handbooks of Economic Development). 1998. [https://books.google.co.uk/books?id=IMG2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Introduction
*Introducing Japan Through Books: A Selected Bibliography. Public Information Bureau, Ministry of Foreign Affairs, Japan. 1968. [https://books.google.co.uk/books?id=FvsyAQAAIAAJ]. 2nd Ed: 1973: [https://books.google.co.uk/books?id=Vj0XAQAAMAAJ].
*Donald Ritchie. Introducing Japan. 1st Ed: 1978. Revised Ed: 1986. 6th printing: 1989: [https://books.google.co.uk/books?id=FE-nxxoKayQC]. 2nd Revised Ed: 1990. 2nd printing: 1991: [https://books.google.co.uk/books?id=hz4UAQAAIAAJ]. 1994: [https://books.google.co.uk/books?id=FMvT6m4SgIQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Webb. An Introduction to Japan. 2nd Ed: 1957: [https://books.google.co.uk/books?id=YQ8MAQAAIAAJ].
*Introducing Modern Japan. A publication of the Japan Information and Culture Center, Embassy of Japan.

Today and yesterday
*Ray Downs. Japan Yesterday and Today. Praeger Publishers. 1970. [https://books.google.co.uk/books?id=PwKxAAAAIAAJ]

Today
*Buckley. Japan Today. 3rd Ed [https://books.google.co.uk/books?id=thyqBtJp2DcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Contemporary
*Routledge Handbook of Contemporary Japan. 2021. [https://books.google.co.uk/books?id=yfH3DwAAQBAJ&amp;pg=PA2011#v=onepage&amp;q&amp;f=false]
*McCargo. Contemporary Japan. 3rd Ed: 2012. [https://books.google.co.uk/books?id=8I5KEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kingston. Contemporary Japan: History, Politics, and Social Change since the 1980s. [https://books.google.co.uk/books?id=enJQZA3R4FMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Modern
*Cortazzi. Modern Japan: A Concise Survey. 1993. [https://books.google.co.uk/books?id=Cf--DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

The Japanese
*Tasker. The Japanese: Portrait of a Nation. 1989 [https://books.google.com/books?id=Q1N8ld78wwQC]
**The Japanese: A Major Exploration of Modern Japan. [https://books.google.co.uk/books?id=CW-6AAAAIAAJ]
**Inside Japan: Wealth, Work and Power in the New Japanese Empire. 1987. [https://books.google.co.uk/books?id=2OJuAAAAMAAJ]

Travel books
*DK Eyewitness Travel: Japan. Reprinted with revisions. 2015: [https://books.google.co.uk/books?id=g2NaBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2017: [https://books.google.co.uk/books?id=vg15DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Dodd and Richmond. The Rough Guide to Japan. 2nd Ed: 2001: [https://books.google.co.uk/books?id=pRGq95ytWZoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Frommer's Japan. 5th Ed: 2000: [https://books.google.co.uk/books?id=-QC8mVyvPa8C].
*Fodor's Japan YYYY. 1984. [https://books.google.co.uk/books?id=aH2Ow27HUQ0C 1986]. [https://books.google.co.uk/books?id=3gTTf6nbv20C 1987]. 1988.
**Fodor's YY Japan. [https://books.google.co.uk/books?id=9QMHllzldlYC 91]. 92. 93.
**Fodor's Japan. 13th Ed: 1996: [https://books.google.co.uk/books?id=cZxZAAAAYAAJ]
*The New Official Guide: Japan. Japan Travel Bureau. 1966. [https://books.google.co.uk/books?id=HoxxAAAAMAAJ]
*Here is Japan. Asahi Broadcasting Corporation. [https://books.google.co.uk/books?id=8QXRCTMNG7MC]
*Japan. (Nagel Travel Guide Series, vol 32). 1964. [https://books.google.co.uk/books?id=QsbXAAAAMAAJ]
*Clark. All the Best in Japan: with Manila, Hong Kong, and Macao. (&quot;All the Best&quot; series). 1959. Reprinted 1964. [https://books.google.co.uk/books?id=yUq4YaaryrwC]. Reviews: [https://archive.dartmouthalumnimagazine.com/article/1958/6/1/all-the-best-in-japan] (1958) 110 Travel 51 [https://books.google.co.uk/books?id=UVwXAQAAMAAJ] 3 Bulletin of the Japan Society, London, No 11: June 1960, p 25 [https://books.google.co.uk/books?id=2oy74hRRXk4C]
**All the Best in Japan and the Orient. 1967.

Music

See [[Universal Bibliography/Music#Japanese and Japan|Music of Japan]]

[[Category:Countries]]</text>
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      <text bytes="36005" sha1="81ndt3q7qp0kf5cehbyk4bbid8lvdbk" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Geography|Geography]].

See [[w:Category:Bibliographies of countries or regions]] and [[w:Category:Works about countries]].

This part of the [[Universal Bibliography]] is a bibliography of countries (including former countries).

==Countries==
*Bateman and Egan (eds). The Encyclopedia of World Geography: A Country by Country Guide. 1993. Revised 1997.
*Peter Stalker. Handbook of the World. 2000. A Guide to Countries of the World. (Oxford Guide to Countries of the World. 2nd Ed: 2004, 2nd Revised Ed: 2007 [https://books.google.co.uk/books?id=GtztAAAAMAAJ], 3rd Ed: 2010 [https://books.google.co.uk/books?id=gvKvfxkbZ1AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Countries of the World and Their Leaders Yearbook. Gale. [https://books.google.co.uk/books?id=5etKAAAAYAAJ] [https://books.google.co.uk/books?id=p41OAAAAIAAJ]
*Hutchinson Guide to Countries of the World [https://books.google.co.uk/books?id=GgpjUe4kN_IC]
*The World Guide: Global Reference, Country by Country. 11th Ed: 2007 [https://books.google.co.uk/books?id=EoWoLgAACAAJ]
*Spence. The World Today: A Nation-by-Nation Guide. Cassell. 1994. 1999. [https://books.google.com/books?id=Ub8qOQAACAAJ]
*Worldmark Encyclopedia of the Nations [https://books.google.co.uk/books?id=I0oYAQAAMAAJ]
*Kurian. Encyclopedia of the World's Nations. Facts on File. Reviews: [https://books.google.co.uk/books?id=Y1EnAQAAIAAJ] [https://books.google.co.uk/books?id=lz0RAQAAMAAJ]
*Michael O'Mara. Facts about the World's Nations. 1999. [https://books.google.co.uk/books?id=mygYAAAAIAAJ]
*Status of the World's Nations. 1965 [https://books.google.co.uk/books?id=sftEyRbAXMUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 1973 [https://books.google.co.uk/books?id=kw2U_Cg2gKYC&amp;pg=PP3#v=onepage&amp;q&amp;f=false].
*[[s:Author:John Alexander Hammerton|Hammerton, John Alexander]] (ed). Countries of the World. Published at the Fleetway House. 6 vols. [https://books.google.co.uk/books?id=e6IaAQAAMAAJ] [https://books.google.co.uk/books?id=K5oaAQAAMAAJ]
*[[s:Author:Robert Brown (1842-1895)|Brown, Robert]]. The Countries of the World. [https://books.google.co.uk/books?id=nO0DAAAAQAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*A Morely Dell. The Countries of the World. (Harrap's New Geographical Series). 1932. (School certificate). Reviews: [https://books.google.co.uk/books?id=oSS9PB_Jf7AC] [https://books.google.co.uk/books?id=BicVAAAAIAAJ] [https://books.google.co.uk/books?id=5qBOAAAAIAAJ] [https://books.google.co.uk/books?id=YbwcAQAAIAAJ] [https://books.google.co.uk/books?id=sc1AAAAAIAAJ]

General series:
*National Geographic Countries of the World [https://books.google.co.uk/books?id=IT2wfzVIPykC]
*Countries of the World. Evans Brothers. (GCSE) [https://books.google.co.uk/books?id=a3sZvWc7E1EC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*One Europe. Longman. [https://search.worldcat.org/en/title/west-germany-adapted-by-lj-russon-from-the-original-german-by-sylvia-lof-ingrid-mallberg-dietrich-rosenthal/oclc/561591761]
*Collier's Nations of the World. The Nations of the World: An Historical Series. [https://books.google.co.uk/books?id=VJY-AAAAYAAJ&amp;pg=PP8#v=onepage&amp;q&amp;f=false]
*Collier's History of Nations. The History of Nations. [https://books.google.co.uk/books?id=fmSUfTY5E80C]
*The Story of the Nations. T Fisher Unwin.
*The World and Its Peoples. (The Illustrated Library of the World and Its Peoples). Greystone Press, New York.
*World and Its Peoples. Marshall Cavendish. [https://books.google.co.uk/books?id=oms5xjI7ba0C&amp;pg=PA141#v=onepage&amp;q&amp;f=false]

==England==
===Counties===
See [[s:Portal:Counties]]

* Harrison, &quot;County Bibliography&quot; (1886) 3 Library Chronicle [https://books.google.co.uk/books?id=Wz9FAAAAYAAJ&amp;pg=PA49#v=onepage&amp;q&amp;f=false 49]

General series
*Victoria County History
*Oxford County Histories
*Pinnock's County Histories
*Shire County Guides. Shire Publications.
*Cambridge County Geographies
*Pike's New Century Series
*[[s:Page:County Churches of Cornwall.djvu/6|County Churches]]. G Allen.

Avon
*Moore. Avon Local History Handbook. Phillimore. 1979. [https://books.google.co.uk/books?id=h0kjAAAAMAAJ] Bibliography, p 102

Bedfordshire
*Conisbee, Lewis Ralph. A Bedfordshire Bibliography. Bedfordshire Historical Record Society. Bedford. 1962. Supplements 1967, 1971, 1978. Third supplement by Threadgill. Review: 6 Archives 52 [https://books.google.co.uk/books?id=oOMZAAAAYAAJ]. See also [https://books.google.co.uk/books?id=MjspAAAAYAAJ] [https://books.google.co.uk/books?id=PejgAAAAMAAJ]
*Godber. History of Bedfordshire. 1969. 1984. [https://books.google.co.uk/books?id=jdvwPQAACAAJ]
*Pinnock. The History and Topography of Bedfordshire [https://books.google.co.uk/books?id=9bJYAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Parry. Select Illustrations, Historical and Topographical, of Bedfordshire [https://books.google.co.uk/books?id=UTUJAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Blyth. The History of Bedford and Visitor's Guide. 1873 [https://books.google.co.uk/books?id=IuIGAAAAQAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Cambridge County Geographies [https://books.google.co.uk/books?id=kTc8AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Buckinghamshire
*Reed. A History of Buckinghamshire. 1993 [https://books.google.co.uk/books?id=BtkWAQAAIAAJ]

Cambridgeshire
*Carter. History of the County of Cambridge [https://books.google.co.uk/books?id=jXpbAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Babington. Ancient Cambridgeshire [https://books.google.co.uk/books?id=DPrCAwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Devon
*Ravenhill and Rowe. Devon Maps and Map-makers [https://books.google.co.uk/books?id=tjf2yAEACAAJ]
*Wright. A Plea for a Devonshire Bibliography. 1885 [https://books.google.co.uk/books?id=8ZUDAAAAQAAJ]

Derbyshire
*Woore. A Catalogue of Local Maps of Derbyshire, C.1528-1800. 2012. [https://books.google.co.uk/books?id=oWmCMwEACAAJ]
*O'Neal. A Bibliography of Derbyshire Lead Mining. 1961

Essex
*Cunnington. Catalogue of Books, Maps and Manuscripts, relating to or connected with the County of Essex. 1902 [https://books.google.co.uk/books?id=oIcqpibGE4MC]
*&quot;The Bibliography of Essex&quot; (1882) 1 Antiquarian Magazine &amp; Bibliographer [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false 72]. See also [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA283#v=onepage&amp;q&amp;f=false p 283].
*&quot;The Bibliography of Essex&quot; (1891) 5 The Essex Naturalist 30 [https://books.google.co.uk/books?id=iIo1AQAAMAAJ]
*Moon. Essex Literature. 1900. Review: 61 Literary World 438 [https://books.google.co.uk/books?id=2T0ZAAAAYAAJ] See also [https://books.google.co.uk/books?id=1Y4UAQAAIAAJ] [https://books.google.co.uk/books?id=C_pEAAAAMAAJ]
*Fenn and Lowery, &quot;An Essex Bibliography&quot;, Journal of the South West Essex Technical College, vols 2 &amp; 3
*Victoria County History bibliography. 1959 [https://books.google.co.uk/books?id=F2EJAQAAIAAJ]
*O'Leary, John Gerard. A Supplement to the Essex Bibliography. Dagenham. 1962.
*A Bibliography of Essex Archaeology &amp; History
*Essex and Dagenham: A Catalogue of Books, Pamphlets and Maps. Dagenham. 1961
*Essex Archaeology and History: The Transactions of the Essex Society for Archaeological and History [https://books.google.co.uk/books?id=CtFAAAAAYAAJ]
*Essex Naturalist: Being the Journal of the Essex Field Club
*Wright. The History and Topography of the County of Essex [https://books.google.co.uk/books?id=SgQVAAAAQAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Ogborne, The History of Essex [https://books.google.co.uk/books?id=IeVSAAAAcAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Suckling. Memorials of the Antiquities and Architecture, Family History and Heraldry of the County of Essex [https://books.google.co.uk/books?id=bcw_AAAAcAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Hunter, The Essex Landscape: A Study of Its Form and History [https://books.google.co.uk/books?id=w9kWAQAAIAAJ]
*Cambridge County Geography [https://books.google.co.uk/books?id=GPHa_X_0qo0C&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Sokoll. Essex  Pauper Letters, 1731-1837 [https://books.google.co.uk/books?id=rCLia7XlqtMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Morant. The History and Antiquities of Colchester in the County of Essex [https://books.google.co.uk/books?id=DDgtAAAAYAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Wallen. The History and Antiquities of the Round Church at Little Maplestead, Essex [https://books.google.co.uk/books?id=FPYVAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Kent
*Smith. Bibliotheca Cantiana. 1837. [https://books.google.co.uk/books?id=1dJDAAAAYAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false]

Leicestershire
*Kirkby, C V (compiler). Catalogue of the books, pamphlets, &amp;c., relating to Leicestershire in the Central Reference Library. Leicester Free Public Libraries. 1893. Reviews: [https://books.google.co.uk/books?id=3boqAQAAIAAJ&amp;pg=PA84#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=UcHnAAAAMAAJ&amp;pg=PA728#v=onepage&amp;q&amp;f=false]
*Leicestershire and Rutland Bibliography, 1963-65 (1966) [https://books.google.co.uk/books?id=-OhVAAAAYAAJ 40] Leicestershire Archaeological and Historical Society: Transactions (1964/5) 92. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1961-63. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1960-61. Available as pdf from University of Leicester.
*A Bibliography of the Small Towns in Leicestershire and Rutland, 1600–1850. (Dissertation). [https://repository.lboro.ac.uk/articles/educational_resource/A_bibliography_of_the_small_towns_in_Leicestershire_and_Rutland_1600_1850/9414200]
*Loughborough's Heritage: A Bibliography of the Holdings of Leicestershire Libraries and Information Service and Record Office. [https://books.google.co.uk/books?id=Bwx2zgEACAAJ]
*Keith Ambrose and Frank Williams, &quot;Bibliography of the Geology of Leicestershire and Rutland: Part 2: 1971-2003&quot; (2004) [https://books.google.co.uk/books?id=U-tQAQAAIAAJ 16] The Mercian Geologist 5. Available as pdf from East Midlands Geological Society.
*Parsons and Brandwood. A Bibliography of Leicestershire Churches. 1978.
*Education in Leicestershire: A Bibliography. [https://books.google.co.uk/books?id=X6EfzQEACAAJ]

Sussex
*Brent, Fletcher and McCann. Sussex in the 16th and 17th Centuries: A Bibliography. 2nd Ed [https://books.google.co.uk/books?id=I7UtAAAAYAAJ]
*Farrant. Sussex in the 18th and 19th Centuries: A Bibliography. 1st Ed: 1973, 2nd Ed: 1977 [https://books.google.co.uk/books?id=MLUtAAAAYAAJ], 3rd Ed: 1979

==France==
Bibliography:
*Bibliographie de la France. Commentary: Encyclopedia of Library and Information Science, vol 37, supplement 2, [https://books.google.co.uk/books?id=10rgjNvOV8oC&amp;pg=PA145#v=onepage&amp;q&amp;f=false p 145]; The Bookseller, 6 January 1881, [https://books.google.co.uk/books?id=4dsiAQAAMAAJ&amp;pg=PA10#v=onepage&amp;q&amp;f=false p 10]; Stein, Manuel de bibliographie générale, [https://books.google.co.uk/books?id=lJYPyKjV1qYC&amp;pg=PA23#v=onepage&amp;q&amp;f=false p 23]. 
*Girault de Saint-Fargeau. Bibliographie historique et topographique de la France. 1845 [https://books.google.co.uk/books?id=kClB9CQNZoMC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Catalogue d'une collection d'ouvrages sur l'histoire des provinces de la France. 1842 [https://books.google.co.uk/books?id=qQBX5WZouzAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Landscape:
*Beaujeu-Garnier. France. (The World's Landscapes). 1975. [https://books.google.com/books?id=nwxDAQAAIAAJ]

Agenais:
*Andrieu. Bibliographie générale de l’Agenais et des parties du Condomois et du Bazadais. 1886 to 1891. Reprinted 1969.

Alsace:
*Ristelhuber. Bibliographie alsacienne. 1869 to 1873 [https://books.google.co.uk/books?id=0mhLAQAAMAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Bibliographie alsacienne: Revue critique des publications concernant l'Alsace. 1918 to 1936
*Ritter. Répertoire bibliographique des livres imprimés en Alsace aux XVe et XVIe siècles [https://books.google.co.uk/books?id=DewaAQAAMAAJ]

Angoumois:
*Castaigne. Essai d'une bibliothèque historique de l'Angoumois, ou Catalogue raisonné des principaux ouvrages qui traitent des différentes branches de l'histoire de cette province. 1847 [https://books.google.co.uk/books?id=R-UanmmlvAEC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Anjou:
*Braguier and Braguier. Archéologie en Anjou: bibliographie. 1984 [https://books.google.co.uk/books?id=LvsmAQAAIAAJ]

Auvergne:
*Gonot. Catalogue des ouvrages imprimés et manuscrits concernant l'Auvergne, extrait du catalogue général de la Bibliotlèque de Clermont-Fd (Puy-de-Dome). 1849. [https://books.google.co.uk/books?id=yCFtbObRCbUC&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Catalogue des livres et estampes concernant l'ancienne Province d'Auvergne (Puy-de-Dôme, Cantal, Haute-Loire) réunis par feu M. G. Desbouis. 1865. [https://books.google.co.uk/books?id=Ui4S8_D0N74C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Béarn
*&quot;Bibliographie Béarnaise&quot;, Revue de Pau et du Béarn [https://books.google.co.uk/books?id=FuZnAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=FQYqvPo9D9IC&amp;pg=PA158#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RL9VAAAAYAAJ]

Brittany
*Sacher. Bibliographie de la Bretagne, ou Catalogue général des ouvrages historiques, littéraires et scientifiques parus sur la Bretagne, avec la liste des revues publiées en cette province, les prix approximatifs des volumes rares, etc. 1881 [https://archive.org/details/bibliographiede00sach]

Burgundy:
*Milsand. Bibliographie bourguignonne; ou, Catalogue méthodique d'ouvrages relatifs à la Bourgogne: Sciences - Arts - Histoire. 1885 [https://archive.org/details/bibliographiebo00milsgoog] [https://archive.org/details/bibliographiebo00sciegoog] [https://books.google.co.uk/books?id=CxIIAAAAQAAJ]
*Catalogue des manuscrits de la Bibliothèque royale des ducs de Bourgogne. 1842 [https://books.google.co.uk/books?id=FX5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*The Companion Guide to Burgundy [https://books.google.co.uk/books?id=NraRP0AkDT0C&amp;pg=PP3#v=onepage&amp;q&amp;f=false]
*Lecat. The Golden Book of Burgundy. (The Golden Book) [https://books.google.co.uk/books?id=FyzR9qU1Zl4C&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Gwynn. Burgundy: With Chapters on the Jura and Savoy. (Kitbag Travel Books). 1935 [https://books.google.co.uk/books?id=ny1LAAAAMAAJ]
*Bazin. Wonderful Burgundy. 1988. 1997 [https://books.google.co.uk/books?id=Yt1CRdICWCUC]
*Bailey. Burgundy. (Insight Guides). 1993 [https://books.google.co.uk/books?id=Q69a1dMW2NQC]
*Dunlop. Burgundy. Hamilton.1990 [https://books.google.co.uk/books?id=S_1OAAAAMAAJ]

Champagne:
*Lhermitte. Ouvrages sur la Champagne: contribution à la bibliographie champenoise. 1992. [https://books.google.co.uk/books?id=jbPfAAAAMAAJ]

Dauphiné:
*Mélanges biographiques et bibliographiques relatifs à l'histoire littéraire du Dauphiné par Colomb de Batines et Ollivier Jules. 1837 [https://books.google.co.uk/books?id=2F5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Lorraine:
*Bibliographie lorraine. Académie nationale de Metz [https://books.google.co.uk/books?id=n-DfAAAAMAAJ]

Maine:
*Desportes. Bibliographie du Maine, précédée de la description topographique et hydrographique du diocése du Mans, Sarthe et Mayenne. 1844. [https://books.google.co.uk/books?id=hSk-AAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Normandy:
*Frère. Manuel du bibliographe Normand ou dictionnaire bibliographique et historique. 1858 to 1860. [https://books.google.co.uk/books?id=dp6geJClg1YC&amp;pg=PP13#v=onepage&amp;q&amp;f=false vol 1]

==Japan==
Bibliography
*Jozef Rogala. A Collector's Guide to Books on Japan in English: An Annotated List of Over 2500 Titles with Subject Index. 2001. [https://books.google.co.uk/books?id=7KI9ao-w2FEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ria Koopmans-de Bruijn. Area Bibliography of Japan. (Scarecrow Area Bibliographies). Scarecrow Press. 1998. [https://books.google.co.uk/books?id=Hlx2OMjgUi0C&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Frank Joseph Shulman. Japan. (World Bibliographical Series, vol 103). Clio Press. 1989. [https://books.google.co.uk/books?id=LsoUAQAAIAAJ]
*Eibun Nihon Kankei Tosho Mokuroku, 1945-1981. (Japanese: 英文日本関係図書目録, 1945-1981). (English: Catalogue of Books in English on Japan, 1945-1981). Japan Foundation. Tokyo. 1986.
*Japan: analytical bibliography: with supplementary research aids: and selected data on Okinawa . . . Department of the Army. Washington. 1972. [https://books.google.co.uk/books?id=h4d4nYxrxtMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Books on Japan in Western Languages. The International Christian University Library. 1971. [https://books.google.co.uk/books?id=F2bQAAAAMAAJ]
*Books on Japan: A List of Acquisitions, 1955-1970. International House of Japan Library. 1971. [https://books.google.co.uk/books?id=F8sWAQAAIAAJ]
*Fukuda. Union Catalog of Books on Japan in Western Languages. 1968. [https://books.google.co.uk/books?id=HKYyAQAAIAAJ]
*A Classified List of Books in Western Languages Relating to Japan. University of Tokyo Press. 1965. [https://books.google.co.uk/books?id=U8MUAQAAIAAJ]
*Katsuji Yabuki (ed). Japan Bibliographic Annual. Published by the Hokuseido Press for the Japan Writers Society. 1956 and 1957.
**Japan Bibliographic Annual 1956. [https://books.google.co.uk/books?id=9XLQAAAAMAAJ]
**Japan Bibliographic Annual 1957. [https://books.google.co.uk/books?id=vesSAAAAIAAJ]. Reviews: (1957) 13 Monumenta Nipponica 166 (April-July) [https://books.google.co.uk/books?id=8S1yb-iwrOwC] (1957) 25 The Oriental Economist 212 (April) [https://books.google.co.uk/books?id=QELoAAAAMAAJ]
*Haring. Books on Japan: A Reference List. 1955. [https://books.google.co.uk/books?id=RbDoAAAAMAAJ]
*Borton. A Selected List of Books and Articles on Japan in English, French, and German. 1940: [https://books.google.co.uk/books?id=YYIsAAAAYAAJ]. Revised and enlarged. Harvard University Press. 1954: [https://books.google.co.uk/books?id=F8O2VwJUPUkC].
**A Selected List of Books on Japan in Western Languages (1945-1960). (Studies on Asia Abroad, vol 1). The Information Centre of Asian Studies, The Toyo Bunko. 1964. [https://books.google.co.uk/books?id=i1_QAAAAMAAJ]
*Oskar Nachod. Bibliography of the Japanese Empire 1906-1926. 1928. [https://archive.org/details/bibliographyofja0001oska/page/n8/mode/1up vol 1]. [https://archive.org/details/bibliographyofja0002oska/page/n6/mode/1up vol 2].
*Fr. von Wenckstern. A Bibliography of the Japanese Empire: being a Classified List of All Books, Essays and Maps in European Languages relating to Dai Nihon (Great Japan) published in Europe, America and in the East from 1859-93 . . . 1895. vol 1. [https://books.google.co.uk/books?id=dcVAAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=v7lO4ddqDywC&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**Volume 2, from 1894 to the middle of 1906. 1907. [https://archive.org/details/bibliographyofja0002frvo/page/n6/mode/1up]
*Hyman Kublin. What Shall I Read on Japan? An Introductory Guide. Japan Society, New York. 1971. [https://books.google.co.uk/books?id=yRRUAAAAYAAJ]

Japanese studies
*An Introductory Bibliography for Japanese Studies. The Japan Foundation. [https://books.google.co.uk/books?id=53O6AAAAIAAJ]
*Richard Perren. Japanese Studies from Pre-History to 1990: A Bibliographical Guide. 1992. [https://books.google.co.uk/books?id=CN9RAQAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies&quot; at pp 1 to 3.
*K.B.S. Bibliography of Standard Reference Books for Japanese Studies, with Descriptive Notes. University of Tokyo Press. [https://books.google.co.uk/books?id=95wbAAAAMAAJ]
*[[w:en:Japan Forum]]. British Association for Japanese Studies. [https://www.tandfonline.com/journals/rjfo20]

History and culture
*John W Dower. Japanese History &amp; Culture from Ancient to Modern Times: Seven Basic Bibliographies. 1986. [https://books.google.co.uk/books?id=NX67AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies &amp; Research Guides&quot; at chapter 6.

Research guides
*Mindy L Kotler. Information Gathering on Japan: A Primer. Search Associates. 1988. ISBN 9780962546006. Catalogue: [https://search.worldcat.org/zh-cn/title/Information-gathering-on-Japan-Joho-:-a-primer/oclc/20530148]. Review: (1989) [https://books.google.co.uk/books?id=NZLiAAAAMAAJ 27] Choice 82

Encyclopedias

See also [[w:ja:Japanese encyclopedias]]
*Louis-Frédéric. Japan Encyclopedia. 2002. [https://books.google.co.uk/books?id=p2QnPijAEmEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Japan: An Illustrated Encyclopedia. Kodansha. 1993.
**Japan: Profile of a Nation. Kodansha. 1995. Revised Edition. 1999.
*[[w:Kodansha Encyclopedia of Japan|Kodansha Encyclopedia of Japan]]. 1983. Supplement. 1986. [https://books.google.co.uk/books?id=WvApAQAAMAAJ]
*Dorothy Perkins. Encyclopedia of Japan: Japanese History and Culture, from Abacus to Zori. Facts on File. A Roundtable Press Book. 1991. [https://books.google.co.uk/books?id=JLKGAAAAIAAJ]
*Pictorial Encyclopedia of Modern Japan. Gakken. 1986. [https://books.google.co.uk/books?id=0FgKAQAAIAAJ]
*Boye Layfayette De Mente. Japan Encyclopedia. 1995. [https://books.google.co.uk/books?id=f9c7AAAAMAAJ]
**Boye De Mente. Everything Japanese. [The Authoritave Reference on Japan Today]. 1989. [https://books.google.co.uk/books?id=Duku89bARgoC]

Media
*[https://www.bbc.com/news/world-asia-pacific-15217593 Japan media guide]. News. BBC. 20 March 2023.
*Masaaki Kasagi. Mass Media in Japan. (Orientation seminars on Japan, number 14). 1983. [https://books.google.co.uk/books?id=odkgAAAAIAAJ]
*Routledge Handbook of Japanese Media [https://books.google.co.uk/books?id=zilKDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Publishers
*[https://www.publishersweekly.com/pw/by-topic/international/international-book-news/article/99729-get-to-know-these-japanese-publishing-companies.html Get to Know These Japanese Publishing Companies]. Publishers Weekly. 20 February 2026.

Press and journalism
*[https://reutersinstitute.politics.ox.ac.uk/digital-news-report/2025/japan Japan]. Reuters Institute for the Study of Journalism. 17 June 2025.
*Marjane Aalam and Philippe Régnier. The Japanese Press and Information System. The Graduate Institute of International Studies. Geneva. [https://books.google.co.uk/books?id=RTcbAQAAIAAJ]
*The Japanese Press: Past and Present. Japan Newspaper Publishers' and Editors' Association. [https://books.google.co.uk/books?id=5tcQAAAAIAAJ 1949].
*Anthony Rausch. Japanese Journalism and the Japanese Newspaper: A Supplemental Reader. [https://books.google.co.uk/books?id=mZrToQEACAAJ]
*Frank L Martin. The Journalism of Japan. 1918. [https://books.google.com/books?id=ruYzAQAAMAAJ]
*William De Lange. A History of Japanese Journalism. Japan Library. 1998. [https://books.google.co.uk/books?id=Rd5tb0cuz8QC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kanesada Hanazono. The Development of Japanese Journalism. Osaka. 1924. [https://books.google.co.uk/books?id=z99ZAAAAMAAJ]
*Kanesada Hanazono. Journalism in Japan and Its Early Pioneers. 1926. [https://books.google.co.uk/books?id=IGTFfLc4bq0C]
*César Castellvi. A Sociology of Journalism in Japan: The Last Empire of the Press. 2024. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PR4#v=onepage&amp;q&amp;f=false]
*&quot;Japan&quot;. Christopher H Sterling (ed). Encyclopedia of Journalism. A Sage Reference Publication. 2009. ISBN 9780761929574. vol 3. pp [https://books.google.co.uk/books?id=ZQhDq8fPj2IC&amp;pg=PA809#v=onepage&amp;q&amp;f=false 809] to 815.

Press annuals
*The Japanese Press. (Nihon Shinbun Kyokai). [https://books.google.co.uk/books?id=AfvyAAAAMAAJ 1979] [https://books.google.co.uk/books?id=Au3yAAAAMAAJ 1998]

Summaries of the press
*Daily Summary of Japanese Press

Foreign correspondents
*Foreign Correspondents in Japan: Reporting a Half Century of Upheavals, from 1945 to the Present. Tuttle. 1998. [https://books.google.co.uk/books?id=YI3TAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Nunn (comp). Japanese Periodicals and Newspapers in Western Languages: An International Union List. Mansell. 1979. [https://books.google.co.uk/books?id=jEROAQAAIAAJ]
*Japan Periodicals. Keizai Koho Center. 3rd Ed [https://books.google.co.uk/books?id=ATm0AAAAIAAJ]. Japan Periodicals, 1982. [https://books.google.co.uk/books?id=PkMyAAAAMAAJ]
*Japanese Periodicals Index
**Humanities and Social Sciences [https://books.google.co.uk/books?id=nXX_RpPGf3AC]
**Natural Sciences [https://books.google.co.uk/books?id=FCJIAAAAYAAJ]
*Current Japanese Periodicals [https://books.google.co.uk/books?id=FjO5AAAAIAAJ]
*Check-list of Japanese Periodicals Held in British University and Research Libraries. [https://books.google.co.uk/books?id=VZgsAAAAYAAJ]
*Union List of Current Japanese Periodicals in the East Asian Libraries of Columbia, Harvard, Princeton, and Yale Universities. [https://books.google.co.uk/books?id=yw7kAAAAMAAJ]
*List of Japanese Periodicals in the Library of the School of Oriental &amp; African Studies. [https://books.google.co.uk/books?id=RREjAQAAIAAJ]
*Gianni Simone. [https://www.japantimes.co.jp/community/2011/04/26/issues/english-mags-approach-milestone-crossroads/ English mags approach milestone, crossroads]. The Japan Times. 26 April 2011.
*Japan Report (1955 onwards) (Consulate General of Japan, Japan Information Center). Vol 39 published in 1993. [https://books.google.co.uk/books?id=MX4BN_frv4IC&amp;pg=PP7#v=onepage&amp;q&amp;f=false] editions:jYuMSMIQC-AC
**Japan Information
*Japan Now [https://books.google.co.uk/books?id=Nul7DRQaexMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Japan Quarterly. (Asahi Shimbun). 1954 to 2001. [https://books.google.co.uk/books?id=nZMMAQAAMAAJ] [https://books.google.co.uk/books?id=_RwVAAAAMAAJ] 189 issues.
*Japan Illustrated: The Japan Times Quarterly [Pictorial] Magazine (October 1963 to Summer 1977) 15 vols [https://books.google.co.uk/books?id=D7UThOmE8T4C]
*[[w:Japan Spotlight|Japan Spotlight]]. Economy, Culture &amp; History: Japan Spotlight: Bimonthly. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ]
*Focus Japan. (Japan External Trade Organization, JETRO). [https://books.google.co.uk/books?id=2fG2hsEZpRkC]
*The Japan Journal [https://books.google.co.uk/books?id=2V3hAAAAMAAJ] [https://books.google.co.uk/books?id=CJwoAQAAMAAJ]
*The Japan Magazine: A Representative Monthly of Things Japanese [https://books.google.co.uk/books?id=ubGKo-p6O_0C] [https://archive.org/details/jm-1914-v4.9-5.2/mode/1up]
*Transactions and Proceedings of the Japan Society, London [https://books.google.co.uk/books?id=B75nnph5qHgC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
**Bulletin. [Bulletin of the Japan Society, London.] [https://books.google.co.uk/books?id=Pd9KvyhnpjMC]
**The Japan Society of London Bulletin [https://books.google.co.uk/books?id=XxlxAAAAMAAJ]
*About Japan. Japan Society, New York. [https://books.google.co.uk/books?id=Nf5OAQAAIAAJ]
**News Bulletin [https://archive.org/details/bub_gb_QcA3AQAAIAAJ/page/n2/mode/1up]
*[[w:en:Metropolis (free magazine)|Metropolis]] (metropolisjapan.com)
*[[w:en:Tokyo Weekender|Tokyo Weekender]] (トーキョー・ウィークエンダー) [https://www.tokyoweekender.com/japan-life/news-and-opinion/nhk-world-features-the-tokyo-weekender-magazine/]
*The Japan Gazette [https://books.google.co.uk/books?id=WSopAAAAYAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Tokio Times [https://books.google.co.uk/books?id=UDfiFBu0vB4C&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*[[w:en:Look Japan|Look Japan]]. (Look Japan Ltd). [https://books.google.co.uk/books?id=QnO6AAAAIAAJ]. Commentary: Gale Directory of Publications and Broadcast Media [https://books.google.co.uk/books?id=ve4dAQAAMAAJ]
*[[w:en:Japan Echo|Japan Echo]]. 1974 to 2010. [https://books.google.co.uk/books?id=Cmq6AAAAIAAJ] [https://books.google.co.uk/books?id=fpmEPpl-85UC]
*PHP Intersect. (Where Japan Meets Asia and the World). PHP Institute. [https://books.google.co.uk/books?id=i74TAQAAMAAJ]
**Intersect Japan [https://books.google.co.uk/books?id=sL8TAQAAMAAJ]
*Speaking of Japan [https://books.google.co.uk/books?id=U7S0AAAAIAAJ]. [Speeches.]
*The Hansei Zasshi: A Monthly Magazine [https://books.google.co.uk/books?id=dyIsvnYjpwEC&amp;pg=PP6#v=onepage&amp;q&amp;f=false]
**The Orient. 1899 onwards [https://books.google.co.uk/books?id=nS1omYYnnd4C&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

Newspapers

See also [[w:List of newspapers in Japan]]
*Tanner. English Language Newspapers in Bakumatsu Japan. 1977. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*[https://www.japantimes.co.jp/news/2009/03/03/reference/newspapers-here-soldiering-on/ Newspapers here soldiering on]. The Japan Times. 3 March 2009.

*[[w:The Japan Times|The Japan Times]]
**The Japan Times: Weekly Edition [https://books.google.co.uk/books?id=KoQ-AQAAMAAJ] [https://books.google.co.uk/books?id=yYQ-AQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Japan Daily Mail
*Japan Weekly Mail
*The Japan Chronicle
**Weekly Edition [https://books.google.co.uk/books?id=vXdRAQAAIAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Japan News. (The Japan News by The Yomiuri Shimbun)
**Yomiuri Japan News (from 1955)
**The Yomiuri (from 1958)
**The Daily Yomiuri (from 1970)
*The Asahi Shimbun: Asia &amp; Japan Watch. [https://www.asahi.com/sp/ajw/]
**Asahi Evening News (from 1954)
***Tokyo Evening News (1952 to 1954) [https://ndlsearch.ndl.go.jp/books/R100000002-I000000145073]
*The Mainichi. [https://mainichi.jp/english/]
**Mainichi Daily News (1922 to 2001) [https://www.nytimes.com/2001/02/27/business/worldbusiness/IHT-tech-briefstop-the-presses.html] [https://ndlsearch.ndl.go.jp/books/R100000002-I000000144910]

Sports newspapers; sports dailies
*Louise do Rosario, &quot;News-stand stars&quot; in &quot;Japan&quot; (1992) [https://books.google.co.uk/books?id=T_GzAAAAIAAJ 155] [[w:en:Far Eastern Economic Review|Far Eastern Economic Review]], 24 to 31 December 1992, p 21
*[[w:ja:岡崎満義|Mitsuyoshi Okazaki]], &quot;Unsportsmanlike Journalism: Japan's sports dailies may be popular, but are they sporting?&quot; in &quot;Sport&quot;, [[w:en:Look Japan|Look Japan]], [https://books.google.co.uk/books?id=lD3tAAAAMAAJ January 1995], p 39

News
*[[w:en:Japan Today|Japan Today]] (ジャパントゥデイ). GPlusMedia. Gakken Holdings.

Annuals and year books 
*This is Japan. Asahi Shimbun. 1954 to 1971. [https://books.google.co.uk/books?id=2X9DAQAAIAAJ]. Commentary: A Victorian Sailor's Grave in the Seto Inland Sea, p 244 [https://books.google.co.uk/books?id=OegkAgAAQBAJ&amp;pg=PA244#v=onepage&amp;q&amp;f=false]
*The Japan Year Book. The Japan Year Book Office. 1905 onwards. [https://archive.org/details/bub_gb_arFPAAAAMAAJ/page/n10/mode/1up 1906]. [https://archive.org/details/in.ernet.dli.2015.553496/page/n27/mode/1up 1915].
*The &quot;Japan Gazette&quot; Japan Year Book. The Japan Gazette. [https://archive.org/details/japan-year-book-1913-1914/page/n15/mode/1up 1913-14]
*The Japan Times Year Book

Almanacs
*Asahi Shimbun Japan Almanac. [https://books.google.co.uk/books?id=SEEEAQAAIAAJ 1995].
*Japan Almanac. (The Mainichi Newspapers). [https://books.google.co.uk/books?id=ufAIAQAAIAAJ 1972]. [https://books.google.co.uk/books?id=X4eXWRkbtFsC 1973]. [https://books.google.co.uk/books?id=7rMrAAAAIAAJ] [https://books.google.co.uk/books?id=krMrAAAAIAAJ]
*[[w:Boyé Lafayette De Mente|Boye De Mente]]. Passport's Japan Almanac. [https://books.google.co.uk/books?id=741wAAAAMAAJ]

General
*Japan: A Country Study. (Area Handbook series). 4th Ed: 1983: [https://books.google.co.uk/books?id=HkM5N3JNc5IC]. 5th Ed: 1992: [https://books.google.co.uk/books?id=ze-wupXxpvEC]
*Area Handbook for Japan. 2nd Ed: 1964: [https://books.google.co.uk/books?id=WucdAAAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]. 3rd Ed: 1974: [https://books.google.co.uk/books?id=LG2aoq1U_eoC&amp;pg=PR1#v=onepage&amp;q&amp;f=false] (DA Pam 550-30).
*Colin Simpson. Picture of Japan.
**Japan: An Intimate View. A S Barnes. [https://books.google.co.uk/books?id=3hkeAAAAMAAJ]
**This is Japan. Angus &amp; Robertson. [https://books.google.co.uk/books?id=HJEJAQAAIAAJ]
*Japan. (The World and Its Peoples). Greystone Press, New York. 1964. Volume 1: [https://books.google.co.uk/books?id=yysUAQAAMAAJ]. Volume 2 &quot;Japan Korea&quot;, including Korea: [https://books.google.co.uk/books?id=uQAUAQAAMAAJ]. See pp 1 to 375 for Japan, and pp 376 to 379 for Ryukyu and Bonin Islands.
*Japan. (World and its Peoples: Eastern and Southern Asia, volume 8). Marshall Cavendish. 2008. ISBN 9780761476412.
*Edward Seidensticker. This Country, Japan. Kodansha International. 1979. ISBN 9780870112294. [https://books.google.co.uk/books?id=88wwAQAAIAAJ]
*Hall and Beardsley. Twelve Doors to Japan. McGraw-Hill. New York. 1965. [https://books.google.co.uk/books?id=0KpxAAAAMAAJ]

Handbooks
*Heenan (ed). The Japan Handbook. (Regional Handbooks of Economic Development). 1998. [https://books.google.co.uk/books?id=IMG2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Introduction
*Introducing Japan Through Books: A Selected Bibliography. Public Information Bureau, Ministry of Foreign Affairs, Japan. 1968. [https://books.google.co.uk/books?id=FvsyAQAAIAAJ]. 2nd Ed: 1973: [https://books.google.co.uk/books?id=Vj0XAQAAMAAJ].
*Donald Ritchie. Introducing Japan. 1st Ed: 1978. Revised Ed: 1986. 6th printing: 1989: [https://books.google.co.uk/books?id=FE-nxxoKayQC]. 2nd Revised Ed: 1990. 2nd printing: 1991: [https://books.google.co.uk/books?id=hz4UAQAAIAAJ]. 1994: [https://books.google.co.uk/books?id=FMvT6m4SgIQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Webb. An Introduction to Japan. 2nd Ed: 1957: [https://books.google.co.uk/books?id=YQ8MAQAAIAAJ].
*Introducing Modern Japan. A publication of the Japan Information and Culture Center, Embassy of Japan.

Today and yesterday
*Ray Downs. Japan Yesterday and Today. Praeger Publishers. 1970. [https://books.google.co.uk/books?id=PwKxAAAAIAAJ]

Today
*Buckley. Japan Today. 3rd Ed [https://books.google.co.uk/books?id=thyqBtJp2DcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Contemporary
*Routledge Handbook of Contemporary Japan. 2021. [https://books.google.co.uk/books?id=yfH3DwAAQBAJ&amp;pg=PA2011#v=onepage&amp;q&amp;f=false]
*McCargo. Contemporary Japan. 3rd Ed: 2012. [https://books.google.co.uk/books?id=8I5KEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kingston. Contemporary Japan: History, Politics, and Social Change since the 1980s. [https://books.google.co.uk/books?id=enJQZA3R4FMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Modern
*Cortazzi. Modern Japan: A Concise Survey. 1993. [https://books.google.co.uk/books?id=Cf--DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

The Japanese
*Tasker. The Japanese: Portrait of a Nation. 1989 [https://books.google.com/books?id=Q1N8ld78wwQC]
**The Japanese: A Major Exploration of Modern Japan. [https://books.google.co.uk/books?id=CW-6AAAAIAAJ]
**Inside Japan: Wealth, Work and Power in the New Japanese Empire. 1987. [https://books.google.co.uk/books?id=2OJuAAAAMAAJ]

Travel books
*DK Eyewitness Travel: Japan. Reprinted with revisions. 2015: [https://books.google.co.uk/books?id=g2NaBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2017: [https://books.google.co.uk/books?id=vg15DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Dodd and Richmond. The Rough Guide to Japan. 2nd Ed: 2001: [https://books.google.co.uk/books?id=pRGq95ytWZoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Frommer's Japan. 5th Ed: 2000: [https://books.google.co.uk/books?id=-QC8mVyvPa8C].
*Fodor's Japan YYYY. 1984. [https://books.google.co.uk/books?id=aH2Ow27HUQ0C 1986]. [https://books.google.co.uk/books?id=3gTTf6nbv20C 1987]. 1988.
**Fodor's YY Japan. [https://books.google.co.uk/books?id=9QMHllzldlYC 91]. 92. 93.
**Fodor's Japan. 13th Ed: 1996: [https://books.google.co.uk/books?id=cZxZAAAAYAAJ]
*The New Official Guide: Japan. Japan Travel Bureau. 1966. [https://books.google.co.uk/books?id=HoxxAAAAMAAJ]
*Here is Japan. Asahi Broadcasting Corporation. [https://books.google.co.uk/books?id=8QXRCTMNG7MC]
*Japan. (Nagel Travel Guide Series, vol 32). 1964. [https://books.google.co.uk/books?id=QsbXAAAAMAAJ]
*Clark. All the Best in Japan: with Manila, Hong Kong, and Macao. (&quot;All the Best&quot; series). 1959. Reprinted 1964. [https://books.google.co.uk/books?id=yUq4YaaryrwC]. Reviews: [https://archive.dartmouthalumnimagazine.com/article/1958/6/1/all-the-best-in-japan] (1958) 110 Travel 51 [https://books.google.co.uk/books?id=UVwXAQAAMAAJ] 3 Bulletin of the Japan Society, London, No 11: June 1960, p 25 [https://books.google.co.uk/books?id=2oy74hRRXk4C]
**All the Best in Japan and the Orient. 1967.

Music

See [[Universal Bibliography/Music#Japanese and Japan|Music of Japan]]

[[Category:Countries]]</text>
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      <text bytes="36082" sha1="mdihsvhxejr46oqxkkqwc3eg08j7v5f" xml:space="preserve">{{Bibliography}}

See also [[Universal Bibliography/Geography|Geography]].

See [[w:Category:Bibliographies of countries or regions]] and [[w:Category:Works about countries]].

This part of the [[Universal Bibliography]] is a bibliography of countries (including former countries).

==Countries==
*Bateman and Egan (eds). The Encyclopedia of World Geography: A Country by Country Guide. 1993. Revised 1997.
*Peter Stalker. Handbook of the World. 2000. A Guide to Countries of the World. (Oxford Guide to Countries of the World. 2nd Ed: 2004, 2nd Revised Ed: 2007 [https://books.google.co.uk/books?id=GtztAAAAMAAJ], 3rd Ed: 2010 [https://books.google.co.uk/books?id=gvKvfxkbZ1AC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Countries of the World and Their Leaders Yearbook. Gale. [https://books.google.co.uk/books?id=5etKAAAAYAAJ] [https://books.google.co.uk/books?id=p41OAAAAIAAJ]
*Hutchinson Guide to Countries of the World [https://books.google.co.uk/books?id=GgpjUe4kN_IC]
*The World Guide: Global Reference, Country by Country. 11th Ed: 2007 [https://books.google.co.uk/books?id=EoWoLgAACAAJ]
*Spence. The World Today: A Nation-by-Nation Guide. Cassell. 1994. 1999. [https://books.google.com/books?id=Ub8qOQAACAAJ]
*Worldmark Encyclopedia of the Nations [https://books.google.co.uk/books?id=I0oYAQAAMAAJ]
*Kurian. Encyclopedia of the World's Nations. Facts on File. Reviews: [https://books.google.co.uk/books?id=Y1EnAQAAIAAJ] [https://books.google.co.uk/books?id=lz0RAQAAMAAJ]
*Michael O'Mara. Facts about the World's Nations. 1999. [https://books.google.co.uk/books?id=mygYAAAAIAAJ]
*Status of the World's Nations. 1965 [https://books.google.co.uk/books?id=sftEyRbAXMUC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 1973 [https://books.google.co.uk/books?id=kw2U_Cg2gKYC&amp;pg=PP3#v=onepage&amp;q&amp;f=false].
*[[s:Author:John Alexander Hammerton|Hammerton, John Alexander]] (ed). Countries of the World. Published at the Fleetway House. 6 vols. [https://books.google.co.uk/books?id=e6IaAQAAMAAJ] [https://books.google.co.uk/books?id=K5oaAQAAMAAJ]
*[[s:Author:Robert Brown (1842-1895)|Brown, Robert]]. The Countries of the World. [https://books.google.co.uk/books?id=nO0DAAAAQAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*A Morely Dell. The Countries of the World. (Harrap's New Geographical Series). 1932. (School certificate). Reviews: [https://books.google.co.uk/books?id=oSS9PB_Jf7AC] [https://books.google.co.uk/books?id=BicVAAAAIAAJ] [https://books.google.co.uk/books?id=5qBOAAAAIAAJ] [https://books.google.co.uk/books?id=YbwcAQAAIAAJ] [https://books.google.co.uk/books?id=sc1AAAAAIAAJ]

General series:
*National Geographic Countries of the World [https://books.google.co.uk/books?id=IT2wfzVIPykC]
*Countries of the World. Evans Brothers. (GCSE) [https://books.google.co.uk/books?id=a3sZvWc7E1EC&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*One Europe. Longman. [https://search.worldcat.org/en/title/west-germany-adapted-by-lj-russon-from-the-original-german-by-sylvia-lof-ingrid-mallberg-dietrich-rosenthal/oclc/561591761]
*Collier's Nations of the World. The Nations of the World: An Historical Series. [https://books.google.co.uk/books?id=VJY-AAAAYAAJ&amp;pg=PP8#v=onepage&amp;q&amp;f=false]
*Collier's History of Nations. The History of Nations. [https://books.google.co.uk/books?id=fmSUfTY5E80C]
*The Story of the Nations. T Fisher Unwin.
*The World and Its Peoples. (The Illustrated Library of the World and Its Peoples). Greystone Press, New York.
*World and Its Peoples. Marshall Cavendish. [https://books.google.co.uk/books?id=oms5xjI7ba0C&amp;pg=PA141#v=onepage&amp;q&amp;f=false]

==England==
===Counties===
See [[s:Portal:Counties]]

* Harrison, &quot;County Bibliography&quot; (1886) 3 Library Chronicle [https://books.google.co.uk/books?id=Wz9FAAAAYAAJ&amp;pg=PA49#v=onepage&amp;q&amp;f=false 49]

General series
*Victoria County History
*Oxford County Histories
*Pinnock's County Histories
*Shire County Guides. Shire Publications.
*Cambridge County Geographies
*Pike's New Century Series
*[[s:Page:County Churches of Cornwall.djvu/6|County Churches]]. G Allen.

Avon
*Moore. Avon Local History Handbook. Phillimore. 1979. [https://books.google.co.uk/books?id=h0kjAAAAMAAJ] Bibliography, p 102

Bedfordshire
*Conisbee, Lewis Ralph. A Bedfordshire Bibliography. Bedfordshire Historical Record Society. Bedford. 1962. Supplements 1967, 1971, 1978. Third supplement by Threadgill. Review: 6 Archives 52 [https://books.google.co.uk/books?id=oOMZAAAAYAAJ]. See also [https://books.google.co.uk/books?id=MjspAAAAYAAJ] [https://books.google.co.uk/books?id=PejgAAAAMAAJ]
*Godber. History of Bedfordshire. 1969. 1984. [https://books.google.co.uk/books?id=jdvwPQAACAAJ]
*Pinnock. The History and Topography of Bedfordshire [https://books.google.co.uk/books?id=9bJYAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Parry. Select Illustrations, Historical and Topographical, of Bedfordshire [https://books.google.co.uk/books?id=UTUJAAAAQAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Blyth. The History of Bedford and Visitor's Guide. 1873 [https://books.google.co.uk/books?id=IuIGAAAAQAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Cambridge County Geographies [https://books.google.co.uk/books?id=kTc8AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Buckinghamshire
*Reed. A History of Buckinghamshire. 1993 [https://books.google.co.uk/books?id=BtkWAQAAIAAJ]

Cambridgeshire
*Carter. History of the County of Cambridge [https://books.google.co.uk/books?id=jXpbAAAAQAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Babington. Ancient Cambridgeshire [https://books.google.co.uk/books?id=DPrCAwAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Devon
*Ravenhill and Rowe. Devon Maps and Map-makers [https://books.google.co.uk/books?id=tjf2yAEACAAJ]
*Wright. A Plea for a Devonshire Bibliography. 1885 [https://books.google.co.uk/books?id=8ZUDAAAAQAAJ]

Derbyshire
*Woore. A Catalogue of Local Maps of Derbyshire, C.1528-1800. 2012. [https://books.google.co.uk/books?id=oWmCMwEACAAJ]
*O'Neal. A Bibliography of Derbyshire Lead Mining. 1961

Essex
*Cunnington. Catalogue of Books, Maps and Manuscripts, relating to or connected with the County of Essex. 1902 [https://books.google.co.uk/books?id=oIcqpibGE4MC]
*&quot;The Bibliography of Essex&quot; (1882) 1 Antiquarian Magazine &amp; Bibliographer [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA72#v=onepage&amp;q&amp;f=false 72]. See also [https://books.google.co.uk/books?id=dEkEAAAAQAAJ&amp;pg=PA283#v=onepage&amp;q&amp;f=false p 283].
*&quot;The Bibliography of Essex&quot; (1891) 5 The Essex Naturalist 30 [https://books.google.co.uk/books?id=iIo1AQAAMAAJ]
*Moon. Essex Literature. 1900. Review: 61 Literary World 438 [https://books.google.co.uk/books?id=2T0ZAAAAYAAJ] See also [https://books.google.co.uk/books?id=1Y4UAQAAIAAJ] [https://books.google.co.uk/books?id=C_pEAAAAMAAJ]
*Fenn and Lowery, &quot;An Essex Bibliography&quot;, Journal of the South West Essex Technical College, vols 2 &amp; 3
*Victoria County History bibliography. 1959 [https://books.google.co.uk/books?id=F2EJAQAAIAAJ]
*O'Leary, John Gerard. A Supplement to the Essex Bibliography. Dagenham. 1962.
*A Bibliography of Essex Archaeology &amp; History
*Essex and Dagenham: A Catalogue of Books, Pamphlets and Maps. Dagenham. 1961
*Essex Archaeology and History: The Transactions of the Essex Society for Archaeological and History [https://books.google.co.uk/books?id=CtFAAAAAYAAJ]
*Essex Naturalist: Being the Journal of the Essex Field Club
*Wright. The History and Topography of the County of Essex [https://books.google.co.uk/books?id=SgQVAAAAQAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Ogborne, The History of Essex [https://books.google.co.uk/books?id=IeVSAAAAcAAJ&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
*Suckling. Memorials of the Antiquities and Architecture, Family History and Heraldry of the County of Essex [https://books.google.co.uk/books?id=bcw_AAAAcAAJ&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Hunter, The Essex Landscape: A Study of Its Form and History [https://books.google.co.uk/books?id=w9kWAQAAIAAJ]
*Cambridge County Geography [https://books.google.co.uk/books?id=GPHa_X_0qo0C&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*Sokoll. Essex  Pauper Letters, 1731-1837 [https://books.google.co.uk/books?id=rCLia7XlqtMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Morant. The History and Antiquities of Colchester in the County of Essex [https://books.google.co.uk/books?id=DDgtAAAAYAAJ&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Wallen. The History and Antiquities of the Round Church at Little Maplestead, Essex [https://books.google.co.uk/books?id=FPYVAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]

Kent
*Smith. Bibliotheca Cantiana. 1837. [https://books.google.co.uk/books?id=1dJDAAAAYAAJ&amp;pg=PP11#v=onepage&amp;q&amp;f=false]

Leicestershire
*Kirkby, C V (compiler). Catalogue of the books, pamphlets, &amp;c., relating to Leicestershire in the Central Reference Library. Leicester Free Public Libraries. 1893. Reviews: [https://books.google.co.uk/books?id=3boqAQAAIAAJ&amp;pg=PA84#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=UcHnAAAAMAAJ&amp;pg=PA728#v=onepage&amp;q&amp;f=false]
*Leicestershire and Rutland Bibliography, 1963-65 (1966) [https://books.google.co.uk/books?id=-OhVAAAAYAAJ 40] Leicestershire Archaeological and Historical Society: Transactions (1964/5) 92. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1961-63. Available as pdf from University of Leicester.
*Leicestershire and Rutland Bibliography, 1960-61. Available as pdf from University of Leicester.
*A Bibliography of the Small Towns in Leicestershire and Rutland, 1600–1850. (Dissertation). [https://repository.lboro.ac.uk/articles/educational_resource/A_bibliography_of_the_small_towns_in_Leicestershire_and_Rutland_1600_1850/9414200]
*Loughborough's Heritage: A Bibliography of the Holdings of Leicestershire Libraries and Information Service and Record Office. [https://books.google.co.uk/books?id=Bwx2zgEACAAJ]
*Keith Ambrose and Frank Williams, &quot;Bibliography of the Geology of Leicestershire and Rutland: Part 2: 1971-2003&quot; (2004) [https://books.google.co.uk/books?id=U-tQAQAAIAAJ 16] The Mercian Geologist 5. Available as pdf from East Midlands Geological Society.
*Parsons and Brandwood. A Bibliography of Leicestershire Churches. 1978.
*Education in Leicestershire: A Bibliography. [https://books.google.co.uk/books?id=X6EfzQEACAAJ]

Sussex
*Brent, Fletcher and McCann. Sussex in the 16th and 17th Centuries: A Bibliography. 2nd Ed [https://books.google.co.uk/books?id=I7UtAAAAYAAJ]
*Farrant. Sussex in the 18th and 19th Centuries: A Bibliography. 1st Ed: 1973, 2nd Ed: 1977 [https://books.google.co.uk/books?id=MLUtAAAAYAAJ], 3rd Ed: 1979

==France==
Bibliography:
*Bibliographie de la France. Commentary: Encyclopedia of Library and Information Science, vol 37, supplement 2, [https://books.google.co.uk/books?id=10rgjNvOV8oC&amp;pg=PA145#v=onepage&amp;q&amp;f=false p 145]; The Bookseller, 6 January 1881, [https://books.google.co.uk/books?id=4dsiAQAAMAAJ&amp;pg=PA10#v=onepage&amp;q&amp;f=false p 10]; Stein, Manuel de bibliographie générale, [https://books.google.co.uk/books?id=lJYPyKjV1qYC&amp;pg=PA23#v=onepage&amp;q&amp;f=false p 23]. 
*Girault de Saint-Fargeau. Bibliographie historique et topographique de la France. 1845 [https://books.google.co.uk/books?id=kClB9CQNZoMC&amp;pg=PP9#v=onepage&amp;q&amp;f=false]
*Catalogue d'une collection d'ouvrages sur l'histoire des provinces de la France. 1842 [https://books.google.co.uk/books?id=qQBX5WZouzAC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Landscape:
*Beaujeu-Garnier. France. (The World's Landscapes). 1975. [https://books.google.com/books?id=nwxDAQAAIAAJ]

Agenais:
*Andrieu. Bibliographie générale de l’Agenais et des parties du Condomois et du Bazadais. 1886 to 1891. Reprinted 1969.

Alsace:
*Ristelhuber. Bibliographie alsacienne. 1869 to 1873 [https://books.google.co.uk/books?id=0mhLAQAAMAAJ&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Bibliographie alsacienne: Revue critique des publications concernant l'Alsace. 1918 to 1936
*Ritter. Répertoire bibliographique des livres imprimés en Alsace aux XVe et XVIe siècles [https://books.google.co.uk/books?id=DewaAQAAMAAJ]

Angoumois:
*Castaigne. Essai d'une bibliothèque historique de l'Angoumois, ou Catalogue raisonné des principaux ouvrages qui traitent des différentes branches de l'histoire de cette province. 1847 [https://books.google.co.uk/books?id=R-UanmmlvAEC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Anjou:
*Braguier and Braguier. Archéologie en Anjou: bibliographie. 1984 [https://books.google.co.uk/books?id=LvsmAQAAIAAJ]

Auvergne:
*Gonot. Catalogue des ouvrages imprimés et manuscrits concernant l'Auvergne, extrait du catalogue général de la Bibliotlèque de Clermont-Fd (Puy-de-Dome). 1849. [https://books.google.co.uk/books?id=yCFtbObRCbUC&amp;pg=PP13#v=onepage&amp;q&amp;f=false]
*Catalogue des livres et estampes concernant l'ancienne Province d'Auvergne (Puy-de-Dôme, Cantal, Haute-Loire) réunis par feu M. G. Desbouis. 1865. [https://books.google.co.uk/books?id=Ui4S8_D0N74C&amp;pg=PP7#v=onepage&amp;q&amp;f=false]

Béarn
*&quot;Bibliographie Béarnaise&quot;, Revue de Pau et du Béarn [https://books.google.co.uk/books?id=FuZnAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=FQYqvPo9D9IC&amp;pg=PA158#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=RL9VAAAAYAAJ]

Brittany
*Sacher. Bibliographie de la Bretagne, ou Catalogue général des ouvrages historiques, littéraires et scientifiques parus sur la Bretagne, avec la liste des revues publiées en cette province, les prix approximatifs des volumes rares, etc. 1881 [https://archive.org/details/bibliographiede00sach]

Burgundy:
*Milsand. Bibliographie bourguignonne; ou, Catalogue méthodique d'ouvrages relatifs à la Bourgogne: Sciences - Arts - Histoire. 1885 [https://archive.org/details/bibliographiebo00milsgoog] [https://archive.org/details/bibliographiebo00sciegoog] [https://books.google.co.uk/books?id=CxIIAAAAQAAJ]
*Catalogue des manuscrits de la Bibliothèque royale des ducs de Bourgogne. 1842 [https://books.google.co.uk/books?id=FX5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
*The Companion Guide to Burgundy [https://books.google.co.uk/books?id=NraRP0AkDT0C&amp;pg=PP3#v=onepage&amp;q&amp;f=false]
*Lecat. The Golden Book of Burgundy. (The Golden Book) [https://books.google.co.uk/books?id=FyzR9qU1Zl4C&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Gwynn. Burgundy: With Chapters on the Jura and Savoy. (Kitbag Travel Books). 1935 [https://books.google.co.uk/books?id=ny1LAAAAMAAJ]
*Bazin. Wonderful Burgundy. 1988. 1997 [https://books.google.co.uk/books?id=Yt1CRdICWCUC]
*Bailey. Burgundy. (Insight Guides). 1993 [https://books.google.co.uk/books?id=Q69a1dMW2NQC]
*Dunlop. Burgundy. Hamilton.1990 [https://books.google.co.uk/books?id=S_1OAAAAMAAJ]

Champagne:
*Lhermitte. Ouvrages sur la Champagne: contribution à la bibliographie champenoise. 1992. [https://books.google.co.uk/books?id=jbPfAAAAMAAJ]

Dauphiné:
*Mélanges biographiques et bibliographiques relatifs à l'histoire littéraire du Dauphiné par Colomb de Batines et Ollivier Jules. 1837 [https://books.google.co.uk/books?id=2F5MAAAAcAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Lorraine:
*Bibliographie lorraine. Académie nationale de Metz [https://books.google.co.uk/books?id=n-DfAAAAMAAJ]

Maine:
*Desportes. Bibliographie du Maine, précédée de la description topographique et hydrographique du diocése du Mans, Sarthe et Mayenne. 1844. [https://books.google.co.uk/books?id=hSk-AAAAYAAJ&amp;pg=PR3#v=onepage&amp;q&amp;f=false]

Normandy:
*Frère. Manuel du bibliographe Normand ou dictionnaire bibliographique et historique. 1858 to 1860. [https://books.google.co.uk/books?id=dp6geJClg1YC&amp;pg=PP13#v=onepage&amp;q&amp;f=false vol 1]

==Japan==
Bibliography
*Jozef Rogala. A Collector's Guide to Books on Japan in English: An Annotated List of Over 2500 Titles with Subject Index. 2001. [https://books.google.co.uk/books?id=7KI9ao-w2FEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Ria Koopmans-de Bruijn. Area Bibliography of Japan. (Scarecrow Area Bibliographies). Scarecrow Press. 1998. [https://books.google.co.uk/books?id=Hlx2OMjgUi0C&amp;pg=PR1#v=onepage&amp;q&amp;f=false]
*Frank Joseph Shulman. Japan. (World Bibliographical Series, vol 103). Clio Press. 1989. [https://books.google.co.uk/books?id=LsoUAQAAIAAJ]
*Eibun Nihon Kankei Tosho Mokuroku, 1945-1981. (Japanese: 英文日本関係図書目録, 1945-1981). (English: Catalogue of Books in English on Japan, 1945-1981). Japan Foundation. Tokyo. 1986.
*Japan: analytical bibliography: with supplementary research aids: and selected data on Okinawa . . . Department of the Army. Washington. 1972. [https://books.google.co.uk/books?id=h4d4nYxrxtMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Books on Japan in Western Languages. The International Christian University Library. 1971. [https://books.google.co.uk/books?id=F2bQAAAAMAAJ]
*Books on Japan: A List of Acquisitions, 1955-1970. International House of Japan Library. 1971. [https://books.google.co.uk/books?id=F8sWAQAAIAAJ]
*Fukuda. Union Catalog of Books on Japan in Western Languages. 1968. [https://books.google.co.uk/books?id=HKYyAQAAIAAJ]
*A Classified List of Books in Western Languages Relating to Japan. University of Tokyo Press. 1965. [https://books.google.co.uk/books?id=U8MUAQAAIAAJ]
*Katsuji Yabuki (ed). Japan Bibliographic Annual. Published by the Hokuseido Press for the Japan Writers Society. 1956 and 1957.
**Japan Bibliographic Annual 1956. [https://books.google.co.uk/books?id=9XLQAAAAMAAJ]
**Japan Bibliographic Annual 1957. [https://books.google.co.uk/books?id=vesSAAAAIAAJ]. Reviews: (1957) 13 Monumenta Nipponica 166 (April-July) [https://books.google.co.uk/books?id=8S1yb-iwrOwC] (1957) 25 The Oriental Economist 212 (April) [https://books.google.co.uk/books?id=QELoAAAAMAAJ]
*Haring. Books on Japan: A Reference List. 1955. [https://books.google.co.uk/books?id=RbDoAAAAMAAJ]
*Borton. A Selected List of Books and Articles on Japan in English, French, and German. 1940: [https://books.google.co.uk/books?id=YYIsAAAAYAAJ]. Revised and enlarged. Harvard University Press. 1954: [https://books.google.co.uk/books?id=F8O2VwJUPUkC].
**A Selected List of Books on Japan in Western Languages (1945-1960). (Studies on Asia Abroad, vol 1). The Information Centre of Asian Studies, The Toyo Bunko. 1964. [https://books.google.co.uk/books?id=i1_QAAAAMAAJ]
*Oskar Nachod. Bibliography of the Japanese Empire 1906-1926. 1928. [https://archive.org/details/bibliographyofja0001oska/page/n8/mode/1up vol 1]. [https://archive.org/details/bibliographyofja0002oska/page/n6/mode/1up vol 2].
*Fr. von Wenckstern. A Bibliography of the Japanese Empire: being a Classified List of All Books, Essays and Maps in European Languages relating to Dai Nihon (Great Japan) published in Europe, America and in the East from 1859-93 . . . 1895. vol 1. [https://books.google.co.uk/books?id=dcVAAAAAYAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false] [https://books.google.co.uk/books?id=v7lO4ddqDywC&amp;pg=PR3#v=onepage&amp;q&amp;f=false]
**Volume 2, from 1894 to the middle of 1906. 1907. [https://archive.org/details/bibliographyofja0002frvo/page/n6/mode/1up]
*Hyman Kublin. What Shall I Read on Japan? An Introductory Guide. Japan Society, New York. 1971. [https://books.google.co.uk/books?id=yRRUAAAAYAAJ]

Japanese studies
*An Introductory Bibliography for Japanese Studies. The Japan Foundation. [https://books.google.co.uk/books?id=53O6AAAAIAAJ]
*Richard Perren. Japanese Studies from Pre-History to 1990: A Bibliographical Guide. 1992. [https://books.google.co.uk/books?id=CN9RAQAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies&quot; at pp 1 to 3.
*K.B.S. Bibliography of Standard Reference Books for Japanese Studies, with Descriptive Notes. University of Tokyo Press. [https://books.google.co.uk/books?id=95wbAAAAMAAJ]
*[[w:en:Japan Forum]]. British Association for Japanese Studies. [https://www.tandfonline.com/journals/rjfo20]

History and culture
*John W Dower. Japanese History &amp; Culture from Ancient to Modern Times: Seven Basic Bibliographies. 1986. [https://books.google.co.uk/books?id=NX67AAAAIAAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. &quot;Bibliographies &amp; Research Guides&quot; at chapter 6.

Research guides
*Mindy L Kotler. Information Gathering on Japan: A Primer. Search Associates. 1988. ISBN 9780962546006. Catalogue: [https://search.worldcat.org/zh-cn/title/Information-gathering-on-Japan-Joho-:-a-primer/oclc/20530148]. Review: (1989) [https://books.google.co.uk/books?id=NZLiAAAAMAAJ 27] Choice 82

Encyclopedias

See also [[w:ja:Japanese encyclopedias]]
*Louis-Frédéric. Japan Encyclopedia. 2002. [https://books.google.co.uk/books?id=p2QnPijAEmEC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Japan: An Illustrated Encyclopedia. Kodansha. 1993.
**Japan: Profile of a Nation. Kodansha. 1995. Revised Edition. 1999.
*[[w:Kodansha Encyclopedia of Japan|Kodansha Encyclopedia of Japan]]. 1983. Supplement. 1986. [https://books.google.co.uk/books?id=WvApAQAAMAAJ]
*Dorothy Perkins. Encyclopedia of Japan: Japanese History and Culture, from Abacus to Zori. Facts on File. A Roundtable Press Book. 1991. [https://books.google.co.uk/books?id=JLKGAAAAIAAJ]
*Pictorial Encyclopedia of Modern Japan. Gakken. 1986. [https://books.google.co.uk/books?id=0FgKAQAAIAAJ]
*Boye Layfayette De Mente. Japan Encyclopedia. 1995. [https://books.google.co.uk/books?id=f9c7AAAAMAAJ]
**Boye De Mente. Everything Japanese. [The Authoritave Reference on Japan Today]. 1989. [https://books.google.co.uk/books?id=Duku89bARgoC]

Media
*[https://www.bbc.com/news/world-asia-pacific-15217593 Japan media guide]. News. BBC. 20 March 2023.
*Masaaki Kasagi. Mass Media in Japan. (Orientation seminars on Japan, number 14). 1983. [https://books.google.co.uk/books?id=odkgAAAAIAAJ]
*Routledge Handbook of Japanese Media [https://books.google.co.uk/books?id=zilKDwAAQBAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]

Publishers
*[https://www.publishersweekly.com/pw/by-topic/international/international-book-news/article/99729-get-to-know-these-japanese-publishing-companies.html Get to Know These Japanese Publishing Companies]. Publishers Weekly. 20 February 2026.

Press and journalism
*[https://reutersinstitute.politics.ox.ac.uk/digital-news-report/2025/japan Japan]. Reuters Institute for the Study of Journalism. 17 June 2025.
*Marjane Aalam and Philippe Régnier. The Japanese Press and Information System. The Graduate Institute of International Studies. Geneva. [https://books.google.co.uk/books?id=RTcbAQAAIAAJ]
*The Japanese Press: Past and Present. Japan Newspaper Publishers' and Editors' Association. [https://books.google.co.uk/books?id=5tcQAAAAIAAJ 1949].
*Anthony Rausch. Japanese Journalism and the Japanese Newspaper: A Supplemental Reader. [https://books.google.co.uk/books?id=mZrToQEACAAJ]
*Frank L Martin. The Journalism of Japan. 1918. [https://books.google.com/books?id=ruYzAQAAMAAJ]
*William De Lange. A History of Japanese Journalism. Japan Library. 1998. [https://books.google.co.uk/books?id=Rd5tb0cuz8QC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kanesada Hanazono. The Development of Japanese Journalism. Osaka. 1924. [https://books.google.co.uk/books?id=z99ZAAAAMAAJ]
*Kanesada Hanazono. Journalism in Japan and Its Early Pioneers. 1926. [https://books.google.co.uk/books?id=IGTFfLc4bq0C]
*César Castellvi. A Sociology of Journalism in Japan: The Last Empire of the Press. 2024. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PR4#v=onepage&amp;q&amp;f=false]
*&quot;Japan&quot;. Christopher H Sterling (ed). Encyclopedia of Journalism. A Sage Reference Publication. 2009. ISBN 9780761929574. vol 3. pp [https://books.google.co.uk/books?id=ZQhDq8fPj2IC&amp;pg=PA809#v=onepage&amp;q&amp;f=false 809] to 815.

Press annuals
*The Japanese Press. (Nihon Shinbun Kyokai). [https://books.google.co.uk/books?id=AfvyAAAAMAAJ 1979] [https://books.google.co.uk/books?id=Au3yAAAAMAAJ 1998]

Summaries of the press
*Daily Summary of Japanese Press

Foreign correspondents
*Foreign Correspondents in Japan: Reporting a Half Century of Upheavals, from 1945 to the Present. Tuttle. 1998. [https://books.google.co.uk/books?id=YI3TAgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Periodicals
*Nunn (comp). Japanese Periodicals and Newspapers in Western Languages: An International Union List. Mansell. 1979. [https://books.google.co.uk/books?id=jEROAQAAIAAJ]
*Japan Periodicals. Keizai Koho Center. 3rd Ed [https://books.google.co.uk/books?id=ATm0AAAAIAAJ]. Japan Periodicals, 1982. [https://books.google.co.uk/books?id=PkMyAAAAMAAJ]
*Japanese Periodicals Index
**Humanities and Social Sciences [https://books.google.co.uk/books?id=nXX_RpPGf3AC]
**Natural Sciences [https://books.google.co.uk/books?id=FCJIAAAAYAAJ]
*Current Japanese Periodicals [https://books.google.co.uk/books?id=FjO5AAAAIAAJ]
*Check-list of Japanese Periodicals Held in British University and Research Libraries. [https://books.google.co.uk/books?id=VZgsAAAAYAAJ]
*Union List of Current Japanese Periodicals in the East Asian Libraries of Columbia, Harvard, Princeton, and Yale Universities. [https://books.google.co.uk/books?id=yw7kAAAAMAAJ]
*List of Japanese Periodicals in the Library of the School of Oriental &amp; African Studies. [https://books.google.co.uk/books?id=RREjAQAAIAAJ]
*Gianni Simone. [https://www.japantimes.co.jp/community/2011/04/26/issues/english-mags-approach-milestone-crossroads/ English mags approach milestone, crossroads]. The Japan Times. 26 April 2011.
*Japan Report (1955 onwards) (Consulate General of Japan, Japan Information Center). Vol 39 published in 1993. [https://books.google.co.uk/books?id=MX4BN_frv4IC&amp;pg=PP7#v=onepage&amp;q&amp;f=false] editions:jYuMSMIQC-AC
**Japan Information
*Japan Now [https://books.google.co.uk/books?id=Nul7DRQaexMC&amp;pg=PP7#v=onepage&amp;q&amp;f=false]
*Japan Quarterly. (Asahi Shimbun). 1954 to 2001. [https://books.google.co.uk/books?id=nZMMAQAAMAAJ] [https://books.google.co.uk/books?id=_RwVAAAAMAAJ] 189 issues.
*Japan Illustrated: The Japan Times Quarterly [Pictorial] Magazine (October 1963 to Summer 1977) 15 vols [https://books.google.co.uk/books?id=D7UThOmE8T4C]
*[[w:Japan Spotlight|Japan Spotlight]]. Economy, Culture &amp; History: Japan Spotlight: Bimonthly. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ]
*Focus Japan. (Japan External Trade Organization, JETRO). [https://books.google.co.uk/books?id=2fG2hsEZpRkC]
*The Japan Journal [https://books.google.co.uk/books?id=2V3hAAAAMAAJ] [https://books.google.co.uk/books?id=CJwoAQAAMAAJ]
*The Japan Magazine: A Representative Monthly of Things Japanese [https://books.google.co.uk/books?id=ubGKo-p6O_0C] [https://archive.org/details/jm-1914-v4.9-5.2/mode/1up]
*Transactions and Proceedings of the Japan Society, London [https://books.google.co.uk/books?id=B75nnph5qHgC&amp;pg=PP5#v=onepage&amp;q&amp;f=false]
**Bulletin. [Bulletin of the Japan Society, London.] [https://books.google.co.uk/books?id=Pd9KvyhnpjMC]
**The Japan Society of London Bulletin [https://books.google.co.uk/books?id=XxlxAAAAMAAJ]
*About Japan. Japan Society, New York. [https://books.google.co.uk/books?id=Nf5OAQAAIAAJ]
**News Bulletin [https://archive.org/details/bub_gb_QcA3AQAAIAAJ/page/n2/mode/1up]
*[[w:en:Metropolis (free magazine)|Metropolis]] (metropolisjapan.com)
*[[w:en:Tokyo Weekender|Tokyo Weekender]] (トーキョー・ウィークエンダー) [https://www.tokyoweekender.com/japan-life/news-and-opinion/nhk-world-features-the-tokyo-weekender-magazine/]
*The Japan Gazette [https://books.google.co.uk/books?id=WSopAAAAYAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Tokio Times [https://books.google.co.uk/books?id=UDfiFBu0vB4C&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*[[w:en:Look Japan|Look Japan]]. (Look Japan Ltd). [https://books.google.co.uk/books?id=QnO6AAAAIAAJ]. Commentary: Gale Directory of Publications and Broadcast Media [https://books.google.co.uk/books?id=ve4dAQAAMAAJ]
*[[w:en:Japan Echo|Japan Echo]]. 1974 to 2010. [https://books.google.co.uk/books?id=Cmq6AAAAIAAJ] [https://books.google.co.uk/books?id=fpmEPpl-85UC]
*PHP Intersect. (Where Japan Meets Asia and the World). PHP Institute. [https://books.google.co.uk/books?id=i74TAQAAMAAJ]
**Intersect Japan [https://books.google.co.uk/books?id=sL8TAQAAMAAJ]
*Speaking of Japan [https://books.google.co.uk/books?id=U7S0AAAAIAAJ]. [Speeches.]
*The Hansei Zasshi: A Monthly Magazine [https://books.google.co.uk/books?id=6qBhfHZo7Q0C&amp;pg=PP5#v=onepage&amp;q&amp;f=false][https://books.google.co.uk/books?id=dyIsvnYjpwEC&amp;pg=PP6#v=onepage&amp;q&amp;f=false]
**The Orient. 1899 onwards [https://books.google.co.uk/books?id=nS1omYYnnd4C&amp;pg=PP5#v=onepage&amp;q&amp;f=false]

Newspapers

See also [[w:List of newspapers in Japan]]
*Tanner. English Language Newspapers in Bakumatsu Japan. 1977. [https://books.google.co.uk/books?id=a2z8EAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*[https://www.japantimes.co.jp/news/2009/03/03/reference/newspapers-here-soldiering-on/ Newspapers here soldiering on]. The Japan Times. 3 March 2009.

*[[w:The Japan Times|The Japan Times]]
**The Japan Times: Weekly Edition [https://books.google.co.uk/books?id=KoQ-AQAAMAAJ] [https://books.google.co.uk/books?id=yYQ-AQAAMAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*Japan Daily Mail
*Japan Weekly Mail
*The Japan Chronicle
**Weekly Edition [https://books.google.co.uk/books?id=vXdRAQAAIAAJ&amp;pg=PA1#v=onepage&amp;q&amp;f=false]
*The Japan News. (The Japan News by The Yomiuri Shimbun)
**Yomiuri Japan News (from 1955)
**The Yomiuri (from 1958)
**The Daily Yomiuri (from 1970)
*The Asahi Shimbun: Asia &amp; Japan Watch. [https://www.asahi.com/sp/ajw/]
**Asahi Evening News (from 1954)
***Tokyo Evening News (1952 to 1954) [https://ndlsearch.ndl.go.jp/books/R100000002-I000000145073]
*The Mainichi. [https://mainichi.jp/english/]
**Mainichi Daily News (1922 to 2001) [https://www.nytimes.com/2001/02/27/business/worldbusiness/IHT-tech-briefstop-the-presses.html] [https://ndlsearch.ndl.go.jp/books/R100000002-I000000144910]

Sports newspapers; sports dailies
*Louise do Rosario, &quot;News-stand stars&quot; in &quot;Japan&quot; (1992) [https://books.google.co.uk/books?id=T_GzAAAAIAAJ 155] [[w:en:Far Eastern Economic Review|Far Eastern Economic Review]], 24 to 31 December 1992, p 21
*[[w:ja:岡崎満義|Mitsuyoshi Okazaki]], &quot;Unsportsmanlike Journalism: Japan's sports dailies may be popular, but are they sporting?&quot; in &quot;Sport&quot;, [[w:en:Look Japan|Look Japan]], [https://books.google.co.uk/books?id=lD3tAAAAMAAJ January 1995], p 39

News
*[[w:en:Japan Today|Japan Today]] (ジャパントゥデイ). GPlusMedia. Gakken Holdings.

Annuals and year books 
*This is Japan. Asahi Shimbun. 1954 to 1971. [https://books.google.co.uk/books?id=2X9DAQAAIAAJ]. Commentary: A Victorian Sailor's Grave in the Seto Inland Sea, p 244 [https://books.google.co.uk/books?id=OegkAgAAQBAJ&amp;pg=PA244#v=onepage&amp;q&amp;f=false]
*The Japan Year Book. The Japan Year Book Office. 1905 onwards. [https://archive.org/details/bub_gb_arFPAAAAMAAJ/page/n10/mode/1up 1906]. [https://archive.org/details/in.ernet.dli.2015.553496/page/n27/mode/1up 1915].
*The &quot;Japan Gazette&quot; Japan Year Book. The Japan Gazette. [https://archive.org/details/japan-year-book-1913-1914/page/n15/mode/1up 1913-14]
*The Japan Times Year Book

Almanacs
*Asahi Shimbun Japan Almanac. [https://books.google.co.uk/books?id=SEEEAQAAIAAJ 1995].
*Japan Almanac. (The Mainichi Newspapers). [https://books.google.co.uk/books?id=ufAIAQAAIAAJ 1972]. [https://books.google.co.uk/books?id=X4eXWRkbtFsC 1973]. [https://books.google.co.uk/books?id=7rMrAAAAIAAJ] [https://books.google.co.uk/books?id=krMrAAAAIAAJ]
*[[w:Boyé Lafayette De Mente|Boye De Mente]]. Passport's Japan Almanac. [https://books.google.co.uk/books?id=741wAAAAMAAJ]

General
*Japan: A Country Study. (Area Handbook series). 4th Ed: 1983: [https://books.google.co.uk/books?id=HkM5N3JNc5IC]. 5th Ed: 1992: [https://books.google.co.uk/books?id=ze-wupXxpvEC]
*Area Handbook for Japan. 2nd Ed: 1964: [https://books.google.co.uk/books?id=WucdAAAAMAAJ&amp;pg=PR1#v=onepage&amp;q&amp;f=false]. 3rd Ed: 1974: [https://books.google.co.uk/books?id=LG2aoq1U_eoC&amp;pg=PR1#v=onepage&amp;q&amp;f=false] (DA Pam 550-30).
*Colin Simpson. Picture of Japan.
**Japan: An Intimate View. A S Barnes. [https://books.google.co.uk/books?id=3hkeAAAAMAAJ]
**This is Japan. Angus &amp; Robertson. [https://books.google.co.uk/books?id=HJEJAQAAIAAJ]
*Japan. (The World and Its Peoples). Greystone Press, New York. 1964. Volume 1: [https://books.google.co.uk/books?id=yysUAQAAMAAJ]. Volume 2 &quot;Japan Korea&quot;, including Korea: [https://books.google.co.uk/books?id=uQAUAQAAMAAJ]. See pp 1 to 375 for Japan, and pp 376 to 379 for Ryukyu and Bonin Islands.
*Japan. (World and its Peoples: Eastern and Southern Asia, volume 8). Marshall Cavendish. 2008. ISBN 9780761476412.
*Edward Seidensticker. This Country, Japan. Kodansha International. 1979. ISBN 9780870112294. [https://books.google.co.uk/books?id=88wwAQAAIAAJ]
*Hall and Beardsley. Twelve Doors to Japan. McGraw-Hill. New York. 1965. [https://books.google.co.uk/books?id=0KpxAAAAMAAJ]

Handbooks
*Heenan (ed). The Japan Handbook. (Regional Handbooks of Economic Development). 1998. [https://books.google.co.uk/books?id=IMG2AgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Introduction
*Introducing Japan Through Books: A Selected Bibliography. Public Information Bureau, Ministry of Foreign Affairs, Japan. 1968. [https://books.google.co.uk/books?id=FvsyAQAAIAAJ]. 2nd Ed: 1973: [https://books.google.co.uk/books?id=Vj0XAQAAMAAJ].
*Donald Ritchie. Introducing Japan. 1st Ed: 1978. Revised Ed: 1986. 6th printing: 1989: [https://books.google.co.uk/books?id=FE-nxxoKayQC]. 2nd Revised Ed: 1990. 2nd printing: 1991: [https://books.google.co.uk/books?id=hz4UAQAAIAAJ]. 1994: [https://books.google.co.uk/books?id=FMvT6m4SgIQC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Webb. An Introduction to Japan. 2nd Ed: 1957: [https://books.google.co.uk/books?id=YQ8MAQAAIAAJ].
*Introducing Modern Japan. A publication of the Japan Information and Culture Center, Embassy of Japan.

Today and yesterday
*Ray Downs. Japan Yesterday and Today. Praeger Publishers. 1970. [https://books.google.co.uk/books?id=PwKxAAAAIAAJ]

Today
*Buckley. Japan Today. 3rd Ed [https://books.google.co.uk/books?id=thyqBtJp2DcC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Contemporary
*Routledge Handbook of Contemporary Japan. 2021. [https://books.google.co.uk/books?id=yfH3DwAAQBAJ&amp;pg=PA2011#v=onepage&amp;q&amp;f=false]
*McCargo. Contemporary Japan. 3rd Ed: 2012. [https://books.google.co.uk/books?id=8I5KEAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]
*Kingston. Contemporary Japan: History, Politics, and Social Change since the 1980s. [https://books.google.co.uk/books?id=enJQZA3R4FMC&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

Modern
*Cortazzi. Modern Japan: A Concise Survey. 1993. [https://books.google.co.uk/books?id=Cf--DAAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]

The Japanese
*Tasker. The Japanese: Portrait of a Nation. 1989 [https://books.google.com/books?id=Q1N8ld78wwQC]
**The Japanese: A Major Exploration of Modern Japan. [https://books.google.co.uk/books?id=CW-6AAAAIAAJ]
**Inside Japan: Wealth, Work and Power in the New Japanese Empire. 1987. [https://books.google.co.uk/books?id=2OJuAAAAMAAJ]

Travel books
*DK Eyewitness Travel: Japan. Reprinted with revisions. 2015: [https://books.google.co.uk/books?id=g2NaBgAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false]. 2017: [https://books.google.co.uk/books?id=vg15DQAAQBAJ&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Dodd and Richmond. The Rough Guide to Japan. 2nd Ed: 2001: [https://books.google.co.uk/books?id=pRGq95ytWZoC&amp;pg=PP1#v=onepage&amp;q&amp;f=false].
*Frommer's Japan. 5th Ed: 2000: [https://books.google.co.uk/books?id=-QC8mVyvPa8C].
*Fodor's Japan YYYY. 1984. [https://books.google.co.uk/books?id=aH2Ow27HUQ0C 1986]. [https://books.google.co.uk/books?id=3gTTf6nbv20C 1987]. 1988.
**Fodor's YY Japan. [https://books.google.co.uk/books?id=9QMHllzldlYC 91]. 92. 93.
**Fodor's Japan. 13th Ed: 1996: [https://books.google.co.uk/books?id=cZxZAAAAYAAJ]
*The New Official Guide: Japan. Japan Travel Bureau. 1966. [https://books.google.co.uk/books?id=HoxxAAAAMAAJ]
*Here is Japan. Asahi Broadcasting Corporation. [https://books.google.co.uk/books?id=8QXRCTMNG7MC]
*Japan. (Nagel Travel Guide Series, vol 32). 1964. [https://books.google.co.uk/books?id=QsbXAAAAMAAJ]
*Clark. All the Best in Japan: with Manila, Hong Kong, and Macao. (&quot;All the Best&quot; series). 1959. Reprinted 1964. [https://books.google.co.uk/books?id=yUq4YaaryrwC]. Reviews: [https://archive.dartmouthalumnimagazine.com/article/1958/6/1/all-the-best-in-japan] (1958) 110 Travel 51 [https://books.google.co.uk/books?id=UVwXAQAAMAAJ] 3 Bulletin of the Japan Society, London, No 11: June 1960, p 25 [https://books.google.co.uk/books?id=2oy74hRRXk4C]
**All the Best in Japan and the Orient. 1967.

Music

See [[Universal Bibliography/Music#Japanese and Japan|Music of Japan]]

[[Category:Countries]]</text>
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      <id>2816367</id>
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      <contributor>
        <username>Platos Cave (physics)</username>
        <id>2562653</id>
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      <text bytes="9540" sha1="2n8wmwzj70wvwfrx28irwln01rsza36" xml:space="preserve">{{Original research}}

:&lt;math&gt;\color{red}ux\color{black} = \frac{v}{r^2} = \frac{1}{k^{1/5} t^{2/15}}&lt;/math&gt; , units = &lt;math&gt;\sqrt{\frac{L}{M T}}&lt;/math&gt; = u&lt;sup&gt;1&lt;/sup&gt; = u (unit number = -13 -15 +30 = 2/2 = 1)

:&lt;math&gt;\color{red}uy\color{black} = k^2t&lt;/math&gt;, units = &lt;math&gt;M^2 T&lt;/math&gt;; (unit number = 15*2 -30 = 0)


{| class=&quot;wikitable&quot;
|+Table 8. Table of Constants (Key:)
! Constant
! Geometry
! θ
! Unit
! Ω&lt;sup&gt;n&lt;/sup&gt;, n = θ − 15 × round(θ / 15)
! CODATA 2014
|-
| Gyromagnetic ratio
| &lt;math&gt;\pi \Omega^3&lt;/math&gt;
| &lt;math&gt;\color{red}-42\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-5} = \frac{t^{3/5}}{k^{8/5}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times -3) \Rightarrow \Omega^3&lt;/math&gt; 
| T = 5.390 517 866 e-44
|-
| Time (Planck)
| &lt;math&gt;\pi&lt;/math&gt;
| &lt;math&gt;\color{red}-30\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-3} = t&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times -2) \Rightarrow 0&lt;/math&gt;
| T = 5.390 517 866 e-44
|-
| Elementary charge  
| &lt;math&gt;\frac{2^7 \pi^4 \Omega^3}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}-27\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-3} = \frac{t^{3/5}}{k^{3/5}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times -2) \Rightarrow \Omega^3&lt;/math&gt;
| ''e&lt;sup&gt;*&lt;/sup&gt;'' = 1.602 176 511 30 e-19
|-
| Length (Planck) 
| &lt;math&gt;2\pi^2 \Omega^2&lt;/math&gt;
| &lt;math&gt;\color{red}-13\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-1} = k^{3/5} t^{{11}/{15}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times -1) \Rightarrow \Omega^2&lt;/math&gt;
| L = 0.161 603 660 096 e-34
|-
| Ampere 
| &lt;math&gt;\frac{2^7 \pi^3 \Omega^3}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}3\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} = \frac{1}{k^{3/5} t^{2/5}} &lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 0) \Rightarrow \Omega^3&lt;/math&gt;
| A = 0.297 221 e25
|-
| Gravitational constant  
| &lt;math&gt;2^3 \pi^4 \Omega^6&lt;/math&gt;
| &lt;math&gt;\color{red}6\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy = k^{4/5} t^{1/5}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 0) \Rightarrow \Omega^6&lt;/math&gt;
| ''G&lt;sup&gt;*&lt;/sup&gt;'' =  6.672 497 192 29 e11
|-
| Mass (Planck)
| &lt;math&gt;1&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}15\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = k&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow 0&lt;/math&gt;
| M =  .217 672 817 580 e-7
|-
| sqrt(momentum)
| &lt;math&gt;\Omega&lt;/math&gt;
| &lt;math&gt;\color{red}16\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{k^{4/5}}{t^{2/{15}}}&lt;/math&gt;
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega&lt;/math&gt;
| 
|-
| Velocity 
| &lt;math&gt;2\pi \Omega^2&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}17\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{k^{3/5}}{t^{4/{15}}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^2&lt;/math&gt;
| V = 299 792 458
|-
| Planck constant  
| &lt;math&gt;2^3 \pi^4 \Omega^4&lt;/math&gt;
| &lt;math&gt;\color{red}19\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^3 = k^{{11}/5} t^{7/{15}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^4&lt;/math&gt;
| ''h&lt;sup&gt;*&lt;/sup&gt;'' = 6.626 069 134 e-34
|-
| Planck temperature  
| &lt;math&gt;\frac{2^7 \pi^3 \Omega^5}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}20\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{1}{t^{2/3}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^5&lt;/math&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  1.418 145 219 e32
|-
| Boltzmann constant
| &lt;math&gt;\frac{a}{2^5 \pi \Omega}&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}29\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^4 = k^{{11}/5} t^{2/{15}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 2) \Rightarrow \Omega^{-1}&lt;/math&gt;
| ''k&lt;sub&gt;B&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 1.379 510 147 52 e-23
|-
| Vacuum permeability 
| &lt;math&gt;\frac{a}{2^{11}\pi^5 \Omega^4}&lt;/math&gt;
| &lt;math&gt;\color{red}56\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^7 = \frac{k^{14/5}}{t^{7/{15}}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 4) \Rightarrow \Omega^{-4}&lt;/math&gt;
| ''μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 4π/10^7
|}


'''Simulation universe modelling at the Planck scale'''

The [[w:simulation hypothesis |simulation hypothesis]] is the proposal that reality could be an artificial simulation, such as a computer simulation. 

The commonly postulated [[w:Ancestor_simulation |ancestor simulation]] approach, which [[w:Nick Bostrom |Nick Bostrom]] called &quot;the simulation argument&quot;, argues for &quot;high-fidelity&quot; simulations of ancestral life that would be indistinguishable from reality to the simulated ancestor. However this simulation variant can be traced back to an 'organic base reality' (the original programmer ancestors and their physical planet). 

The Programmer God hypothesis conversely states that a (deep universe) simulation began with the big bang and was programmed by an external intelligence (external to the physical universe), the Programmer by definition a God in the creator of the universe context. Our universe in its entirety, down to the smallest detail, and including life-forms, is within the simulation, the Laws of Nature, at their most fundamental level, are coded rules running on top of the simulation Operating System. 

The &quot;high-fidelity&quot; simulation requires only that the observable region of space be simulated (as with computer games), conversely the theoretically observable region of a deep-universe simulation would extend to the [[w:Planck_units |Planck scale]] (beyond this scale the Laws of Physics break down).


===Constraints===
Any candidate for a Programmer-God source code must satisfy these conditions;
# It can generate physical structures from mathematical forms.
# The sum universe is dimensionless (simply data on a celestial hard disk).
# We must be able to use it to derive the laws of physics (it is the origin of the laws of nature).
# The mathematical logic must be unknown to us (the Programmer is a non-human intelligence).
# The coding should demonstrate an 'elegance' commensurate with the Programmer's level of skill.


===Feasibility===
While the philosophy of an ancestor simulation can be extrapolated using existing computer science modelling as a reference, decoding the Programmer God source code requires extrapolating from the laws of physics. The following is a list of pages that discuss the physics (and mathematics) of a Programmer God Simulation hypothesis under these 5 given constraints. The criteria for inclusion being that no dimensioned constants can be fundamental (see condition #2), each page is restricted to the dimensionless [[w:fine structure constant |fine structure constant]] alpha, the mathematical constants pi and e, and an incrementally expanding universe (the origin of integers). As pi and e can be derived from integers in series, they may not be considered as necessary initial constants. Furthermore a specific geometrical artifice may be used as the ''guard-rail''. 

For AI analysis, the model has been compiled into a single pdf (https://codingthecosmos.com/Programmer-God_Simulation-Hypothesis_Complete-Model-2025.pdf). 


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God (programmer)]]: An introduction to a Planck scale universe. Argues that the OS for the universe would operate at the Planck scale and would employ geometrical objects rather than numbers to construct the Planck units. Demonstrates how to construct the physical units (kg, m, s, A) from mathematical structures.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical)]]: The electron as a geometrical formula for which embedded within are the geometrical objects MTLVPA ... it is these objects which confer the physical electron properties, the electron itself is a mathematical, not physical particle. The base-15 rules permit an electron quark configuration DDD, however a positron quark configuration would then have to be DUU (equivalent to the proton).


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Physical_constant_(anomaly)]]: Anomalies within the physical constants (G, h, c, e, m&lt;sub&gt;e&lt;/sub&gt; k&lt;sub&gt;B&lt;/sub&gt;) as evidence of a Simulation Universe
 

* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity emerges from geometric averaging of N-body rotating orbital pairs, reproducing Kepler's laws and Mercury perihelion precession without forces. Hydrogen spectroscopy emerges from hyperbolic spiral geometry with transition frequencies matching experiment to 0.001% using only alpha, pi and Compton wavelengths. Quantum numbers &lt;math&gt;(n, l, m_l, m_s)&lt;/math&gt; are geometric properties of nested helical paths in 4D spacetime. 


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units as geometrical objects from the mathematical constants pi and e (M = 1, T = pi, ...) with only Planck charge using alpha. The unit relationship between the objects is contrained by a geometrical base-15.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: The universe expands in discrete Planck unit increments, with Cosmic Microwave Background parameters derivable from the Spiral of Theodorus geometry. Despite using only Planck units, the parameters are within 6% of the established CMB values.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Motion originates from a 4-axis hypersphere expansion at velocity &lt;math&gt;c = l_p/t_p&lt;/math&gt;, with 3D space residing on the hypersphere surface. Relativity emerges as coordinate transformation between absolute hypersphere expansion and relative 3D motion.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Momentum unit as the link between mass and charge


==References==
{{Reflist}}


[[Category:Physics| ]]
[[Category:Philosophy of science| ]]</text>
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    <revision>
      <id>2816372</id>
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        <username>Platos Cave (physics)</username>
        <id>2562653</id>
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      <text bytes="10267" sha1="bwqqvaxwdq0vm8t244i26ac43cbv5wy" xml:space="preserve">{{Original research}}


:&lt;math&gt;\color{red}ux\color{black} = \frac{v}{r^2} = \frac{1}{k^{1/5} t^{2/15}}&lt;/math&gt; , units = &lt;math&gt;\sqrt{\frac{L}{M T}}&lt;/math&gt; = u&lt;sup&gt;1&lt;/sup&gt; = u (unit number = -13 -15 +30 = 2/2 = 1)

:&lt;math&gt;\color{red}uy\color{black} = k^2t&lt;/math&gt;, units = &lt;math&gt;M^2 T&lt;/math&gt;; (unit number = 15*2 -30 = 0)

:&lt;math&gt;\color{red}uA\color{black}&lt;/math&gt;, units = &lt;math&gt;\frac{L^{3/2}}{M^{3/2} T^{3/2}}&lt;/math&gt;; (unit number = 3)

:&lt;math&gt;\color{red}f(x)\color{black}&lt;/math&gt;, units = &lt;math&gt;\frac{L^{15}}{M^9 T^{11}}&lt;/math&gt;; (unit number = 0)


{| class=&quot;wikitable&quot;
|+Table 8. Table of Constants (Key:)
! Constant
! Geometry
! θ
! Unit
! Ω&lt;sup&gt;n&lt;/sup&gt;, n = θ − 15 × round(θ / 15)
! CODATA 2014
|-
| Gyromagnetic ratio
| &lt;math&gt;\pi \Omega^3&lt;/math&gt;
| &lt;math&gt;\color{red}-42\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-5} = \frac{t^{3/5}}{k^{8/5}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times -3) \Rightarrow \Omega^3&lt;/math&gt; 
| &lt;math&gt;\frac{ux^\theta f(x)^3}{uy^5} = \frac{u_A  T}{M}\; (\frac{A  s}{kg})&lt;/math&gt;
|-
| Time (Planck)
| &lt;math&gt;\pi&lt;/math&gt;
| &lt;math&gt;\color{red}-30\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-3} = t&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times -2) \Rightarrow 0&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta f(x)^2}{uy^3} = T\;(s)&lt;/math&gt;
|-
| Elementary charge  
| &lt;math&gt;\frac{2^7 \pi^4 \Omega^3}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}-27\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-3} = \frac{t^{3/5}}{k^{3/5}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times -2) \Rightarrow \Omega^3&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta f(x)^2}{uy^3} = u_A T \;(A s)&lt;/math&gt;
|-
| Length (Planck) 
| &lt;math&gt;2\pi^2 \Omega^2&lt;/math&gt;
| &lt;math&gt;\color{red}-13\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-1} = k^{3/5} t^{{11}/{15}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times -1) \Rightarrow \Omega^2&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta f(x)}{uy} = L\;(m)&lt;/math&gt;
|-
| Ampere 
| &lt;math&gt;\frac{2^7 \pi^3 \Omega^3}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}3\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} = \frac{1}{k^{3/5} t^{2/5}} &lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 0) \Rightarrow \Omega^3&lt;/math&gt;
| &lt;math&gt;ux^\theta = u_A\; (A)&lt;/math&gt;
|-
| Gravitational constant  
| &lt;math&gt;2^3 \pi^4 \Omega^6&lt;/math&gt;
| &lt;math&gt;\color{red}6\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy = k^{4/5} t^{1/5}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 0) \Rightarrow \Omega^6&lt;/math&gt;
| &lt;math&gt;ux^{\theta} uy = \frac{L^3}{M T^2}\; (\frac{m^3}{kg s^2})&lt;/math&gt;
|-
| Mass (Planck)
| &lt;math&gt;1&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}15\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = k&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow 0&lt;/math&gt;
| &lt;math&gt;\frac{ux^{\theta} uy^2}{f(x)} = M\; (kg)&lt;/math&gt;
|-
| sqrt(momentum)
| &lt;math&gt;\Omega&lt;/math&gt;
| &lt;math&gt;\color{red}16\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{k^{4/5}}{t^{2/{15}}}&lt;/math&gt;
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega&lt;/math&gt;
| &lt;math&gt;\frac{ux^{\theta} uy^2}{f(x)} = \sqrt{\frac{M L}{T}}\; (\sqrt{\frac{kg m}{s}})&lt;/math&gt; 
|-
| Velocity 
| &lt;math&gt;2\pi \Omega^2&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}17\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{k^{3/5}}{t^{4/{15}}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^2&lt;/math&gt;
| &lt;math&gt;\frac{ux^{\theta} uy^2}{f(x)} = \frac{L}{T}\; (\frac{m}{s})&lt;/math&gt;
|-
| Planck constant  
| &lt;math&gt;2^3 \pi^4 \Omega^4&lt;/math&gt;
| &lt;math&gt;\color{red}19\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^3 = k^{{11}/5} t^{7/{15}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^4&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^3}{f(x)} = \frac{M L^2}{T}\; (\frac{kg m^2}{s})&lt;/math&gt;
|-
| Planck temperature  
| &lt;math&gt;\frac{2^7 \pi^3 \Omega^5}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}20\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{1}{t^{2/3}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^5&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^2}{f(x)} = \frac{u_A L}{T}\; (\frac{A m}{s})&lt;/math&gt;
|-
| Boltzmann constant
| &lt;math&gt;\frac{a}{2^5 \pi \Omega}&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}29\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^4 = k^{{11}/5} t^{2/{15}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 2) \Rightarrow \Omega^{-1}&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^4}{f(x)^2} = \sqrt{\frac{M^5 T}{L}}\; (\frac{kg m}{A s})&lt;/math&gt;
|-
| Vacuum permeability 
| &lt;math&gt;\frac{a}{2^{11}\pi^5 \Omega^4}&lt;/math&gt;
| &lt;math&gt;\color{red}56\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^7 = \frac{k^{14/5}}{t^{7/{15}}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 4) \Rightarrow \Omega^{-4}&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^7}{f(x)^4} = \frac{M^4 T}{L^2}\; (\frac{kg m}{A^2 s^2})&lt;/math&gt;
|}


'''Simulation universe modelling at the Planck scale'''

The [[w:simulation hypothesis |simulation hypothesis]] is the proposal that reality could be an artificial simulation, such as a computer simulation. 

The commonly postulated [[w:Ancestor_simulation |ancestor simulation]] approach, which [[w:Nick Bostrom |Nick Bostrom]] called &quot;the simulation argument&quot;, argues for &quot;high-fidelity&quot; simulations of ancestral life that would be indistinguishable from reality to the simulated ancestor. However this simulation variant can be traced back to an 'organic base reality' (the original programmer ancestors and their physical planet). 

The Programmer God hypothesis conversely states that a (deep universe) simulation began with the big bang and was programmed by an external intelligence (external to the physical universe), the Programmer by definition a God in the creator of the universe context. Our universe in its entirety, down to the smallest detail, and including life-forms, is within the simulation, the Laws of Nature, at their most fundamental level, are coded rules running on top of the simulation Operating System. 

The &quot;high-fidelity&quot; simulation requires only that the observable region of space be simulated (as with computer games), conversely the theoretically observable region of a deep-universe simulation would extend to the [[w:Planck_units |Planck scale]] (beyond this scale the Laws of Physics break down).


===Constraints===
Any candidate for a Programmer-God source code must satisfy these conditions;
# It can generate physical structures from mathematical forms.
# The sum universe is dimensionless (simply data on a celestial hard disk).
# We must be able to use it to derive the laws of physics (it is the origin of the laws of nature).
# The mathematical logic must be unknown to us (the Programmer is a non-human intelligence).
# The coding should demonstrate an 'elegance' commensurate with the Programmer's level of skill.


===Feasibility===
While the philosophy of an ancestor simulation can be extrapolated using existing computer science modelling as a reference, decoding the Programmer God source code requires extrapolating from the laws of physics. The following is a list of pages that discuss the physics (and mathematics) of a Programmer God Simulation hypothesis under these 5 given constraints. The criteria for inclusion being that no dimensioned constants can be fundamental (see condition #2), each page is restricted to the dimensionless [[w:fine structure constant |fine structure constant]] alpha, the mathematical constants pi and e, and an incrementally expanding universe (the origin of integers). As pi and e can be derived from integers in series, they may not be considered as necessary initial constants. Furthermore a specific geometrical artifice may be used as the ''guard-rail''. 

For AI analysis, the model has been compiled into a single pdf (https://codingthecosmos.com/Programmer-God_Simulation-Hypothesis_Complete-Model-2025.pdf). 


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God (programmer)]]: An introduction to a Planck scale universe. Argues that the OS for the universe would operate at the Planck scale and would employ geometrical objects rather than numbers to construct the Planck units. Demonstrates how to construct the physical units (kg, m, s, A) from mathematical structures.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical)]]: The electron as a geometrical formula for which embedded within are the geometrical objects MTLVPA ... it is these objects which confer the physical electron properties, the electron itself is a mathematical, not physical particle. The base-15 rules permit an electron quark configuration DDD, however a positron quark configuration would then have to be DUU (equivalent to the proton).


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Physical_constant_(anomaly)]]: Anomalies within the physical constants (G, h, c, e, m&lt;sub&gt;e&lt;/sub&gt; k&lt;sub&gt;B&lt;/sub&gt;) as evidence of a Simulation Universe
 

* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity emerges from geometric averaging of N-body rotating orbital pairs, reproducing Kepler's laws and Mercury perihelion precession without forces. Hydrogen spectroscopy emerges from hyperbolic spiral geometry with transition frequencies matching experiment to 0.001% using only alpha, pi and Compton wavelengths. Quantum numbers &lt;math&gt;(n, l, m_l, m_s)&lt;/math&gt; are geometric properties of nested helical paths in 4D spacetime. 


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units as geometrical objects from the mathematical constants pi and e (M = 1, T = pi, ...) with only Planck charge using alpha. The unit relationship between the objects is contrained by a geometrical base-15.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: The universe expands in discrete Planck unit increments, with Cosmic Microwave Background parameters derivable from the Spiral of Theodorus geometry. Despite using only Planck units, the parameters are within 6% of the established CMB values.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Motion originates from a 4-axis hypersphere expansion at velocity &lt;math&gt;c = l_p/t_p&lt;/math&gt;, with 3D space residing on the hypersphere surface. Relativity emerges as coordinate transformation between absolute hypersphere expansion and relative 3D motion.


* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Momentum unit as the link between mass and charge


==References==
{{Reflist}}


[[Category:Physics| ]]
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'''Natural Planck units as geometrical objects (the mathematical electron model)'''

The physical constants form the scaffolding around which the theories of physics are erected, and they define the fabric of our universe, but science has no idea why they take the special numerical values that they do, for these constants follow no discernible pattern. The desire to explain the constants has been one of the driving forces behind efforts to develop a complete unified description of nature, or &quot;theory of everything&quot;. Physicists have hoped that such a theory would show that each of the constants of nature ''could have only one logically possible value''. It would reveal an underlying order to the seeming arbitrariness of nature &lt;ref&gt;J. Barrow, J. Webb {{Cite journal |title= Inconsistent constants |journal=Scientific American |volume=292 |pages=56 |date=2005}}&lt;/ref&gt;.


In the [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical) |mathematical electron]] &lt;ref&gt;Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}&lt;/ref&gt; model, the electron is assigned a geometrical formula ψ, the formula itself the geometry of 2 dimensionless constants (α, Ω), and resembles the formula for the volume of a torus or surface of a 4-axis hypersphere ψ = 4π&lt;sup&gt;2&lt;/sup&gt;r&lt;sup&gt;3&lt;/sup&gt;. 


Embedded within this formula ψ are geometrical analogues of the [[w:Planck units |Planck units]] mass M, time T, length L, ampere A. It is these MLTA Planck objects which confer the electron properties (mass, wavelength, charge ...), and the magnitude of these properties is determined directly by the formula ψ. Thus this geometrical formula for the electron itself encodes the information required to produce the observed physical electron. For example, we can write;

[[w:electron mass | electron mass]]     &lt;math&gt;m_e = \frac{M}{\psi}&lt;/math&gt;

[[w:Compton wavelength | electron wavelength]] &lt;math&gt;\lambda_e = 2\pi L \psi&lt;/math&gt;


=== Planck unit limitations ===
The SI Planck units are measured; [[w:Planck mass |Planck mass]] in ''kg'', [[w:Planck length |Planck length]] in ''m'', [[w:Planck time |Planck time]] in ''s'' ... . These units have numerical values, the problem then becomes to derive a mathematical relation between these SI units, because for this we cannot use numerical values; numerical values are simply dimensionless frequencies of the SI unit itself, 299792458 could refer to the speed of light 299792458m/s or equally to the number of apples in a container (299792458 apples), numbers such as 299792458 carry no unit-specific information, and so the units are treated as independent by default. This therefore requires that to the number 299792458 is added a descriptive (the unit), which could be m/s or apples.


=== Geometrical objects ===
This inherent restriction can be resolved by assigning to each unit a geometrical object MLTA for which the geometry embeds the attribute (for example, the geometry of the time object T embeds the function time and so a descriptive unit ''s'' = seconds is not required). We may then combine these objects Lego-style to form more complex objects; from electrons to galaxies, while still retaining the underlying attributes (of mass M, wavelength L, frequency T ...). An apple has mass because its 'geometry' includes the geometrical object for mass.

There are 2 principal Planck objects required; mass M = 1 and time T = π, also there is a physical constant ([[w:fine-structure constant | fine structure constant '''α''']]), and a mathematical constant Omega. Omega itself is the geometry of π and Euler's number [[w:E_(mathematical_constant) |e]] = 2.718281828459...;

:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.007\;134\;9543... &lt;/math&gt;


A physical constant is defined as a constant embedded within the universe &quot;source code&quot; (it is a given). A mathematical constant however can be derived within the boundaries of the universe itself; π and e can be constructed from integers in series. We can offer an averaged fine structure constant alpha, here assigned the letter ''a'' = 137.03599... to represent the analogue of the inverse fine structure constant α&lt;sup&gt;-1&lt;/sup&gt; = 137.03599...

In order to combine geometrical objects we require a mathematical relationship between them, here denoted by a unit number θ
{| class=&quot;wikitable&quot;
|+Table 1. MT fundamental Planck objects
! attribute
! geometrical object
! unit number θ
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;
| 15
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;
| -30
|}


From MTΩα we can derive further Planck unit analogues momentum P, length L and charge A.  
{| class=&quot;wikitable&quot;
|+Table 2. PLTVA Geometrical objects
! attribute
! geometrical object
! unit number θ
|-
| [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]
| &lt;math&gt;P = (\Omega) M^{(4/5)} \;(\pi/T)^{(2/15)} = \Omega&lt;/math&gt;
| 16
|-
| velocity  
| &lt;math&gt;V = \frac{2\pi P^2}{M} = (2\pi\Omega^2)&lt;/math&gt;
| 17
|-
| length  
| &lt;math&gt;L = VT = (2\pi^2\Omega^2)&lt;/math&gt; 
| -13
|-
| ampere 
| &lt;math&gt;A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| 3
|-
| temperature 
| &lt;math&gt;K = \frac{AV}{2\pi} = (\frac{2^7 \pi^3 \Omega^5}{a})&lt;/math&gt;
| 20
|}


As the geometries of dimensionless constants, these objects are also dimensionless and so are independent of any system of units, and of any numerical system, and so could qualify as &quot;natural units&quot; (naturally occuring units);
{{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...''
...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as &quot;natural units&quot;... -Max Planck &lt;ref&gt;Planck (1899), p. 479.&lt;/ref&gt;&lt;ref name=&quot;TOM&quot;&gt;*Tomilin, K. A., 1999, &quot;[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]&quot;, 287–296.&lt;/ref&gt;}}  


=== Scalars ===

To translate from geometrical objects to a numerical system of units requires system dependent scalars ('''kltpva'''). For example; 

:If we use ''k'' to convert ''M'' (M=1) to the SI Planck mass (M*''k''&lt;sub&gt;SI&lt;/sub&gt; = &lt;math&gt;m_P&lt;/math&gt;), then ''k''&lt;sub&gt;SI&lt;/sub&gt; = 0.2176728e-7kg ([[w:SI_units |SI units]])

:''c'' = V*''v''&lt;sub&gt;SI&lt;/sub&gt; = 299792458m/s ([[w:SI_units |SI units]])

:''c'' = V*''v''&lt;sub&gt;imp&lt;/sub&gt; = 186282miles/s ([[w:Imperial_units |imperial units]])


==== Scalar relationships ====
Scalars that translate to the SI unit system must therefore carry not only the numerical conversion but also the unit, i.e.: scalar ''v'' = 11843707.905 m/s. This also means that the scalars follow the unit number relationship θ which we can use in our formulas as ''u''&lt;sup&gt;θ&lt;/sup&gt;. 

{| class=&quot;wikitable&quot;
|+Table 3. Geometrical units
! Attribute
! Geometrical object
! Scalar 
! Unit ''u''&lt;sup&gt;θ&lt;/sup&gt;
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;  
| ''k''
| &lt;math&gt;u^{15}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;  
| ''t''
| &lt;math&gt;u^{-30}&lt;/math&gt;
|-
| [[v:User:Platos Cave (physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]
| &lt;math&gt;P = (\Omega)&lt;/math&gt;
| ''r''&lt;sup&gt;2&lt;/sup&gt;
| &lt;math&gt;u^{16}&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = (2\pi\Omega^2)&lt;/math&gt;  
| ''v''
| &lt;math&gt;u^{17}&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = (2\pi^2\Omega^2)&lt;/math&gt; 
| ''l''
| &lt;math&gt;u^{-13}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| ''q''
| &lt;math&gt;u^3&lt;/math&gt;
|}


Here are examples where units = 1, as such ''only 2 scalars are required'', for example, if we know the numerical value for ''q'' and for ''l'' then we know the numerical value for ''t'' ('''t = q&lt;sup&gt;3&lt;/sup&gt;l&lt;sup&gt;3&lt;/sup&gt;'''), and from ''l'' and ''t'' we know the value for ''k''. 

:&lt;math&gt;\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{q^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1&lt;/math&gt;


In other words, once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, [[w:permeability of vacuum|μ&lt;sub&gt;0&lt;/sub&gt;]]) are assigned exact values, following the [[w:2019 redefinition of SI base units|2019 redefinition of SI base units]] a total of 4 constants now have independently exact values assigned which is problematic in terms of this model. 


Scalars ''r'' (θ = 8) and ''v'' (θ = 17) are chosen here for demonstration as they can be derived directly from the 2 constants with exact values; ''c'' and ''μ&lt;sub&gt;0&lt;/sub&gt;''.
:&lt;math&gt;c = 299792458&lt;/math&gt; m/s
:&lt;math&gt;\mu_0 = 4\pi / 10^7&lt;/math&gt;
:&lt;math&gt;v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = \frac{m}{s}&lt;/math&gt;
:&lt;math&gt;r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{a};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;

{| class=&quot;wikitable&quot;
|+Table 4. Geometrical objects
! attribute
! geometrical object
! unit number θ
! scalar r(8), v(17)
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt;
| 15 = 8*4-17
| &lt;math&gt;k = \frac{r^4}{v}&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt;
|  -30 = 8*9-17*6 
| &lt;math&gt;t = \frac{r^9}{v^6}&lt;/math&gt;
|-
| sqrt(momentum)  
| &lt;math&gt;P = (\Omega) M^{(4/5)} \;(\pi/T)^{(2/15)} = \Omega&lt;/math&gt;
| 17
| &lt;math&gt;r^2&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = \frac{2\pi P^2}{M} = (2\pi\Omega^2)&lt;/math&gt;
| 17
| &lt;math&gt;v&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = VT = (2\pi^2\Omega^2)&lt;/math&gt; 
| -13 = 8*9-17*5
| &lt;math&gt;l = \frac{r^9}{v^5}&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| 3 = 17*3-8*6
| &lt;math&gt;\frac{v^3}{r^6}&lt;/math&gt;
|}


{| class=&quot;wikitable&quot;
|+ Table 5. Comparison; SI and θ
! constant
! θ (SI unit)
! MLTVA
! scalar r(8), v(17)
|-
| ''c'' 
| &lt;math&gt;\frac{m}{s}&lt;/math&gt; (-13+30 = {{font color|red|white|17}})
| ''c*'' = &lt;math&gt;V*v&lt;/math&gt; 
| {{font color|red|white|17}}
|-
| ''h'' 
| &lt;math&gt;\frac{kg \;m^2}{s}&lt;/math&gt; (15-26+30={{font color|red|white|19}})
| ''h*'' = &lt;math&gt;2 \pi M V L * \frac{r^{13}}{v^5}&lt;/math&gt; 
| 8*13-17*5={{font color|red|white|19}}
|-
| ''G''
| &lt;math&gt;\frac{m^3}{kg \;s^2}&lt;/math&gt; (-39-15+60={{font color|red|white|6}})
| ''G*'' = &lt;math&gt;\frac{V^2 L}{M} * \frac{r^5}{v^2}&lt;/math&gt; 
| 8*5-17*2={{font color|red|white|6}}
|-
| ''e'' 
| &lt;math&gt;C = A s&lt;/math&gt; (3-30={{font color|red|white|-27}})
| ''e*'' = &lt;math&gt;A T * \frac{r^3}{v^3}&lt;/math&gt; 
| 8*3-17*3={{font color|red|white|-27}}
|-
| ''k&lt;sub&gt;B&lt;/sub&gt;'' 
| &lt;math&gt;\frac{kg \;m^2}{s^2 \;K}&lt;/math&gt; (15-26+60-20={{font color|red|white|29}})
| ''k&lt;sub&gt;B&lt;/sub&gt;*'' = &lt;math&gt;\frac{2 \pi V M}{A} * \frac{r^{10}}{v^3}&lt;/math&gt; 
| 8*10-17*3={{font color|red|white|29}}
|-
| ''μ&lt;sub&gt;0&lt;/sub&gt;'' 
| &lt;math&gt;\frac{kg \;m}{s^2 \;A^2}&lt;/math&gt; (15-13+60-6={{font color|red|white|56}})
| ''μ&lt;sub&gt;0&lt;/sub&gt;*'' = &lt;math&gt;\frac{4 \pi V^2 M}{a L A^2} * r^7&lt;/math&gt; 
| 8*7={{font color|red|white|56}}
|}


====CODATA 2014====
Following the 26th General Conference on Weights and Measures ([[w:2019 redefinition of SI base units|2019 redefinition of SI base units]]) are fixed the numerical values of the 4 physical constants (''h, c, e, k&lt;sub&gt;B&lt;/sub&gt;''), consequently here we are using CODATA 2014 values. This is because only 2 dimensioned physical constants can be assigned exact values, once 2 constants (2 scalars) have been assigned values, then all other constants are defined by default. In CODATA 2014 2 constants have exact values; &lt;math&gt;c&lt;/math&gt; and the [[w:Vacuum permeability | vacuum permeability]] &lt;math&gt;\mu_0&lt;/math&gt;. 


=== Dimensionless f(x) ===
From our unit number relationship we can build a generic dimensionless formula f&lt;sub&gt;X&lt;/sub&gt;;

&lt;math&gt;f_X = \frac{kg^9 s^{11}}{m^{15}} = \frac{(\frac{r^4}{v})^9 (\frac{r^9}{v^6})^{11}}{(\frac{r^9}{v^5})^{15}} = 1&lt;/math&gt;

This f&lt;sub&gt;X&lt;/sub&gt;, although embedded within are the dimensioned structures for mass, time and length (in the above ratio), would be a dimensionless mathematical structure, units = 1. Thus we may create as much mass, time and length as we wish, the only proviso being that they are created in f&lt;sub&gt;X&lt;/sub&gt; ratios, so that regardless of how massive, old and large our universe becomes, it is still in sum total dimensionless. 

Defining the dimensioned quantities ''r'', ''v'' in SI unit terms.

:&lt;math&gt;r = (\frac{kg\;m}{s})^{1/4}&lt;/math&gt;

:&lt;math&gt;v = \frac{m}{s}&lt;/math&gt;


Mass
:&lt;math&gt;\frac{r^4}{v} = (\frac{kg\;m}{s})\;(\frac{s}{m}) = kg&lt;/math&gt;


Length
:&lt;math&gt;(r^9)^4 = \frac{kg^9\;m^9}{s^9} &lt;/math&gt;

:&lt;math&gt;(\frac{1}{v^5})^4 = \frac{s^{20}}{m^{20}}&lt;/math&gt;

:&lt;math&gt;(\frac{r^9}{v^5})^4 = \frac{kg^9 s^{11}}{m^{11}} = m^4 \frac{kg^9 s^{11}}{m^{15}} = m^4 f_X = m^4&lt;/math&gt;


Time
:&lt;math&gt;(r^9)^4 = \frac{kg^9\;m^9}{s^9} &lt;/math&gt;

:&lt;math&gt;(\frac{1}{v^6})^4 = \frac{s^{24}}{m^{24}}&lt;/math&gt;

:&lt;math&gt;(\frac{r^9}{v^6})^4 = \frac{kg^9 s^{15}}{m^{15}} = s^4 \frac{kg^9 s^{11}}{m^{15}} = s^4 f_X = s^4&lt;/math&gt;


And so, although f&lt;sub&gt;X&lt;/sub&gt; is a dimensionless mathematical structure, we can embed within it the (mass, length, time ...) structures along with their dimensional attributes (kg, m, s, A ..). The electron itself is an example of an f&lt;sub&gt;X&lt;/sub&gt; structure, it (f&lt;sub&gt;electron&lt;/sub&gt;) is a dimensionless geometrical object that embeds the physical electron parameters of wavelength, frequency, charge (note: A-m = ampere-meter are the units for a [[w:Magnetic_monopole#In_SI_units |magnetic monopole]]). 

&lt;math&gt;f_{electron}&lt;/math&gt; 

:&lt;math&gt;units = \frac{A^3 m^3}{s} = \frac{(\frac{v^3}{r^6})^3 (\frac{r^9}{v^5})^3}{(\frac{r^9}{v^6})} = 1&lt;/math&gt;  


We may note that at the macro-level (of planets and stars) these f&lt;sub&gt;X&lt;/sub&gt; ratio are not found, and so this level is the domain of the observed physical universe, however at the quantum level, f&lt;sub&gt;X&lt;/sub&gt; ratio do appear, f&lt;sub&gt;electron&lt;/sub&gt; as an example, the mathematical and physical domains then blurring. This would also explain why physics can measure precisely the parameters of the electron (wavelength, mass ...), but has never found the electron itself.


=== Fine structure constant ===
Classically the fine structure constant can be expressed by this formula.

:&lt;math&gt;\frac{2 h}{\mu_0 e^2 c} = \color{red}\alpha^{-1} \color{black}&lt;/math&gt;


If we insert the geometrical analogues in this formula then alpha emerges, units and scalars cancel, serving to validate the unit number relationship and the geometries.

:&lt;math&gt;\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{a}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{a})^2(2 \pi \Omega^2) = 
 \color{red}a \color{black}&lt;/math&gt;

:&lt;math&gt;units \;\frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1&lt;/math&gt;

:&lt;math&gt;scalars \;(\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1&lt;/math&gt;

Thus proving that &lt;math&gt; \color{red}\alpha\color{black} = \color{red}\alpha \color{black}&lt;/math&gt;


=== Electron formula ===
{{main|User:Platos Cave (physics)/Simulation_Hypothesis/Electron (mathematical)}}
The ''electron object'' (''formula ψ'') is a mathematical particle (units and scalars cancel). 

:&lt;math&gt;\psi = 4\pi^2(2^6 3 \pi^2 a \Omega^5)^3 = .23895453...x10^{23}&lt;/math&gt; units = 1


In this example, embedded within the electron are the objects for charge, length and time ALT. AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]].

:&lt;math&gt;T = \pi \frac{r^9}{v^6},\; u^{-30}&lt;/math&gt;

:&lt;math&gt;\sigma_{e} = \frac{3 a^2 A L}{2\pi^2} = {2^7 3 \pi^3 a \Omega^5}\frac{r^3}{v^2},\; u^{-10}&lt;/math&gt;

:&lt;math&gt;\psi = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 a \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1&lt;/math&gt;


Associated with the electron are dimensioned parameters, these parameters however are a function of the MLTA units, the formula ψ dictating the frequency of these units.

[[w:electron mass | electron mass]]     &lt;math&gt;m_e^* = \frac{M}{\psi}&lt;/math&gt; (M = [[w:Planck mass | Planck mass]] = &lt;math&gt;\frac{r^4}{v})&lt;/math&gt; 

[[w:Compton wavelength | electron wavelength]] &lt;math&gt;\lambda_e^* = 2\pi L \psi&lt;/math&gt; (L = [[w:Planck length | Planck length]] = &lt;math&gt;2\pi\Omega^2\frac{r^9}{v^5})&lt;/math&gt;

[[w:elementary charge | elementary charge]]    &lt;math&gt;e^* = A\;T&lt;/math&gt; (T = [[w:Planck time | Planck time]]) = &lt;math&gt;\frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3}&lt;/math&gt; 

[[w:Rydberg constant | Rydberg constant]] &lt;math&gt;R^* = (\frac{m_e}{4 \pi L a^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} a^5 \Omega^{17}}\frac{v^5}{r^9}\;u^{13}&lt;/math&gt; 


=== Omega ===
There is a natural number solution for Ω that is a square root implying that Ω can have a plus or a minus solution, and this agrees with the requirements of this theory (in the mass domain Ω occurs as Ω&lt;sup&gt;2&lt;/sup&gt; = plus only, in the charge domain Ω occurs as Ω&lt;sup&gt;3&lt;/sup&gt; = can be plus or minus; see [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum | sqrt(momentum)]]). 

:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.007\;134\;9543... &lt;/math&gt;

We may also consider including [[w:Euler%27s_formula | Euler's_formula]] where {{mvar|i}} is the [[w:imaginary unit | imaginary unit]] &lt;ref&gt;http://simulationuniverse.org/files/Omega-derivation-final.pdf Derivation of Omega&lt;/ref&gt;. 


=== Dimensionless combinations ===
According to the unit number relationship, we can also combine the physical constants in combinations where the unit numbers cancel, in this model these combinations are dimensionless, however they still retain SI units. If the model is correct (if the combinations are dimensionless) then the scalars will also have cancelled and numerically the solutions using CODATA or Geometrical objects will approach equality (barring uncertainties). These combinations can be used to test the veracity of the MLTA geometries as natural Planck units.


Example:

:&lt;math&gt;\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = &lt;/math&gt; {{font color|green|yellow| '''0.228 473 759... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}

:&lt;math&gt;\frac{h^3}{e^{13} c^{24}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 639... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}, units = &lt;math&gt;\frac{kg^3 s^8}{m^{18} A^{13}}&lt;/math&gt;, units = 1 (15*3-30*8+13*18-3*13 = 0)

 Note: the geometry &lt;math&gt;\color{red}(\Omega^{15})^n\color{black}&lt;/math&gt; (integer n ≥ 0) is common to all ratios where units and scalars cancel (i.e.: only combinations with &lt;math&gt;\Omega^0, \Omega^{15}, \Omega^{30}, \Omega^{45}&lt;/math&gt;... will be dimensionless). However there is no Planck unit with a &lt;math&gt;\Omega^{15}&lt;/math&gt; component (all constants are combinations of &lt;math&gt;\Omega^2&lt;/math&gt; and &lt;math&gt;\Omega^3&lt;/math&gt;), and this suggests there is an underlying geometrical base-15.


{| class=&quot;wikitable&quot;
|+Table 6. Dimensionless combinations
! CODATA 2014 mean
! (α, Ω) mean
! units = 1
! scalars = 1
|-
| &lt;math&gt;\frac{k_B e c}{h} =&lt;/math&gt; {{font color|green|yellow|'''1.000 8254'''}}
| &lt;math&gt;\frac{(k_B^*) (e^*) (c^*)}{(h^*)}&lt;/math&gt; = {{font color|green|yellow|'''1.0'''}}
| &lt;math&gt;\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1&lt;/math&gt;
|-
| &lt;math&gt;\frac{h^3}{e^{13} c^{24}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 639... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} =&lt;/math&gt; {{font color|green|yellow| '''0.228 473 759... 10&lt;sup&gt;-58&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1&lt;/math&gt;
|-
| &lt;math&gt;\frac{c^9 e^4}{m_e^3} =&lt;/math&gt; {{font color|green|yellow| '''0.170 514 342... 10&lt;sup&gt;92&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black}=&lt;/math&gt; {{font color|green|yellow| '''0.170 514 368... 10&lt;sup&gt;92&lt;/sup&gt;'''}}
| &lt;math&gt;\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1&lt;/math&gt;
| &lt;math&gt;(v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1&lt;/math&gt; 
|-
| &lt;math&gt;\frac{k_B}{e^2 m_e c^4} =&lt;/math&gt; {{font color|green|yellow| '''73 095 507 858.'''}}
| &lt;math&gt;\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} =&lt;/math&gt; {{font color|green|yellow| '''73 035 235 897.'''}} 
| &lt;math&gt;\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1&lt;/math&gt;
| &lt;math&gt;(\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1&lt;/math&gt;
|}


=== Derivation via CODATA ===
In this section, we show how to numerically solve the least precise dimensioned physical constants (''G'', ''h'', ''e'', ''m''&lt;sub&gt;e&lt;/sub&gt;, ''k''&lt;sub&gt;B&lt;/sub&gt; ...) in terms of the 3 most precise dimensioned physical constants); [[w:Speed of light | speed of light]] '''c''' (exact value), [[w:Vacuum permeability | vacuum permeability]] '''μ&lt;sub&gt;0&lt;/sub&gt;''' (exact value), [[w:Rydberg constant | Rydberg constant]] '''R''' (12-13 digits). 


We first look for combinations in which the unit numbers are equal, and then add dimensionless numbers as required. For example;
:&lt;math&gt;{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{3^{19} \pi^{12} \Omega^{12} r^{39}u^{57}}{v^{15}},\; \theta = 57&lt;/math&gt;

:&lt;math&gt;\frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2} = \frac{3^{19} \pi^{12} \Omega^{12} r^{39} u^{57}}{v^{15}},\; \theta = 57&lt;/math&gt;


We then replace the geometrical with the SI (''c'', ''μ&lt;sub&gt;0&lt;/sub&gt;'', ''R'') 

&lt;math&gt;{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 a^{13} {R}^2}&lt;/math&gt;


{| class=&quot;wikitable&quot;
|+Table 7. R, c, μ&lt;sub&gt;0&lt;/sub&gt;, a ... 
! constant
! formula*
! θ
! Units
|-
| [[w:Planck constant | Planck constant]]
| &lt;math&gt;{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 a^{13} {R}^2}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3}{A^6 s}&lt;/math&gt;, 15*3-3*6+30 = {{font color|red|white|57}}
| &lt;math&gt;\frac{kg \;m^2}{s}&lt;/math&gt;, θ = 15-13*2+30 = {{font color|red|white|19}}
|-
| [[w:Gravitational constant | Gravitational constant]]
| &lt;math&gt;{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 a^{11} {R}^2}&lt;/math&gt;
| &lt;math&gt;\frac{kg\; m^3}{A^2 s^2}&lt;/math&gt;, 15-13*3-3*2+30*2 = {{font color|red|white|30}}
| &lt;math&gt;\frac{m^3}{kg \;s^2}&lt;/math&gt;, θ = -13*3-15+30*2 = {{font color|red|white|6}}
|-
| [[w:Elementary charge | Elementary charge]]
| &lt;math&gt;{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 a^8 {R}}&lt;/math&gt;
| &lt;math&gt;\frac{s^3}{m^3}&lt;/math&gt;, -30*4+13*3 = {{font color|red|white|-81}}
| &lt;math&gt;A s&lt;/math&gt;, θ = 3-30 = {{font color|red|white|-27}}
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| &lt;math&gt;{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 a^5 {R}}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3}{s^2 A^6}&lt;/math&gt;, 15*3+30*2-3*6 = {{font color|red|white|87}}
| &lt;math&gt;\frac{kg \;m^2}{s^2 \;K}&lt;/math&gt;, θ = 15-26+60-20 = {{font color|red|white|29}}
|-
| [[w:Electron mass | Electron mass]]
| &lt;math&gt;{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 a^7}&lt;/math&gt;
| &lt;math&gt;\frac{kg^3 s^2}{m^6 A^6}&lt;/math&gt;, 15*3-30*2+13*6-3*6 = {{font color|red|white|45}}
| &lt;math&gt;kg&lt;/math&gt;, θ = {{font color|red|white|15}}
|-
| [[w:Planck length | Planck length]]
| &lt;math&gt;({l_p^*})^{15} = \frac{\pi^{22} {\mu_0}^9}{2^{35} 3^{24} a^{49} c^{35} R^8}&lt;/math&gt;
| &lt;math&gt;\frac{kg^9 s^{17}}{m^{18}A^{18}}&lt;/math&gt;, 15*9-30*17+13*18-3*18 = {{font color|red|white|-195}}
| &lt;math&gt;m&lt;/math&gt;, θ = {{font color|red|white|-13}}
|-
| [[w:Planck mass | Planck mass]]
| &lt;math&gt;({m_P^*})^{15} = \frac{2^{25} \pi^{13} {\mu_0}^6}{3^6 c^5 a^{16} R^2}&lt;/math&gt;
| u = &lt;math&gt;\frac{kg^6 m^3}{s^7 A^{12}}&lt;/math&gt;, 15*6-13*3+30*7-3*12 = {{font color|red|white|225}}
| &lt;math&gt;kg&lt;/math&gt;, θ = {{font color|red|white|15}}
|}


=== Base-15 geometry ===
:&lt;math&gt;\color{red}ux\color{black} = \frac{v}{r^2} = \frac{1}{k^{1/5} t^{2/15}}&lt;/math&gt; , units = &lt;math&gt;\sqrt{\frac{L}{M T}}&lt;/math&gt; = u&lt;sup&gt;1&lt;/sup&gt; = u (unit number = -13 -15 +30 = 2/2 = 1)

:&lt;math&gt;\color{red}uy\color{black} = k^2t&lt;/math&gt;, units = &lt;math&gt;M^2 T&lt;/math&gt;; (unit number = 15*2 -30 = 0)

:&lt;math&gt;\color{red}uA\color{black}&lt;/math&gt;, units = &lt;math&gt;\frac{L^{3/2}}{M^{3/2} T^{3/2}}&lt;/math&gt;; (unit number = 3)

:&lt;math&gt;\color{red}f(x)\color{black}&lt;/math&gt;, units = &lt;math&gt;\sqrt{\frac{L^{15}}{M^9 T^{11}}}&lt;/math&gt;; (unit number = 0)

:&lt;math&gt;\color{red}k\color{black} = 0.21767282521 \times 10^{-7} &lt;/math&gt;M (kg)
:&lt;math&gt;\color{red}t\color{black} = 0.1715855294 \times 10^{-43} &lt;/math&gt;T (s)
:&lt;math&gt;\color{red}k^2 t\color{black} = 0.81299726963 \times 10^{-59} &lt;/math&gt; M&lt;sup&gt;2&lt;/sup&gt;T (kg&lt;sup&gt;2&lt;/sup&gt;s)
:&lt;math&gt;\color{red}\Omega\color{black} = 2.007134954324946&lt;/math&gt;


{| class=&quot;wikitable&quot;
|+Table 8. Table of Constants (Key:)
! Constant
! Geometry
! θ
! Unit
! Ω&lt;sup&gt;n&lt;/sup&gt;, n = θ − 15 × round(θ / 15)
! SI equivalent
|-
| Gyromagnetic ratio
| &lt;math&gt;\pi \Omega^3&lt;/math&gt;
| &lt;math&gt;\color{red}-42\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-5} = \frac{t^{3/5}}{k^{8/5}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times -3) \Rightarrow \Omega^3&lt;/math&gt; 
| &lt;math&gt;\frac{ux^\theta f(x)^3}{uy^5} = \frac{u_A  T}{M}\; (\frac{A  s}{kg})&lt;/math&gt;
|-
| Time (Planck)
| &lt;math&gt;\pi&lt;/math&gt;
| &lt;math&gt;\color{red}-30\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-3} = t&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times -2) \Rightarrow 0&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta f(x)^2}{uy^3} = T\;(s)&lt;/math&gt;
|-
| Elementary charge  
| &lt;math&gt;\frac{2^7 \pi^4 \Omega^3}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}-27\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-3} = \frac{t^{3/5}}{k^{3/5}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times -2) \Rightarrow \Omega^3&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta f(x)^2}{uy^3} = u_A T \;(A s)&lt;/math&gt;
|-
| Length (Planck) 
| &lt;math&gt;2\pi^2 \Omega^2&lt;/math&gt;
| &lt;math&gt;\color{red}-13\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^{-1} = k^{3/5} t^{{11}/{15}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times -1) \Rightarrow \Omega^2&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta f(x)}{uy} = L\;(m)&lt;/math&gt;
|-
| Ampere 
| &lt;math&gt;\frac{2^7 \pi^3 \Omega^3}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}3\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} = \frac{1}{k^{3/5} t^{2/5}} &lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 0) \Rightarrow \Omega^3&lt;/math&gt;
| &lt;math&gt;ux^\theta = u_A\; (A)&lt;/math&gt;
|-
| Gravitational constant  
| &lt;math&gt;2^3 \pi^4 \Omega^6&lt;/math&gt;
| &lt;math&gt;\color{red}6\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy = k^{4/5} t^{1/5}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 0) \Rightarrow \Omega^6&lt;/math&gt;
| &lt;math&gt;ux^{\theta} uy = \frac{L^3}{M T^2}\; (\frac{m^3}{kg s^2})&lt;/math&gt;
|-
| Mass (Planck)
| &lt;math&gt;1&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}15\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = k&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow 0&lt;/math&gt;
| &lt;math&gt;\frac{ux^{\theta} uy^2}{f(x)} = M\; (kg)&lt;/math&gt;
|-
| sqrt(momentum)
| &lt;math&gt;\Omega&lt;/math&gt;
| &lt;math&gt;\color{red}16\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{k^{4/5}}{t^{2/{15}}}&lt;/math&gt;
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega&lt;/math&gt;
| &lt;math&gt;\frac{ux^{\theta} uy^2}{f(x)} = \sqrt{\frac{M L}{T}}\; (\sqrt{\frac{kg m}{s}})&lt;/math&gt; 
|-
| Velocity 
| &lt;math&gt;2\pi \Omega^2&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}17\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{k^{3/5}}{t^{4/{15}}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^2&lt;/math&gt;
| &lt;math&gt;\frac{ux^{\theta} uy^2}{f(x)} = \frac{L}{T}\; (\frac{m}{s})&lt;/math&gt;
|-
| Planck constant  
| &lt;math&gt;2^3 \pi^4 \Omega^4&lt;/math&gt;
| &lt;math&gt;\color{red}19\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^3 = k^{{11}/5} t^{7/{15}}&lt;/math&gt; 
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^4&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^3}{f(x)} = \frac{M L^2}{T}\; (\frac{kg m^2}{s})&lt;/math&gt;
|-
| Planck temperature  
| &lt;math&gt;\frac{2^7 \pi^3 \Omega^5}{a}&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}20\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^2 = \frac{1}{t^{2/3}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 1) \Rightarrow \Omega^5&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^2}{f(x)} = \frac{u_A L}{T}\; (\frac{A m}{s})&lt;/math&gt;
|-
| Boltzmann constant
| &lt;math&gt;\frac{a}{2^5 \pi \Omega}&lt;/math&gt;
| &lt;math&gt;\color{red}\color{red}29\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^4 = k^{{11}/5} t^{2/{15}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 2) \Rightarrow \Omega^{-1}&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^4}{f(x)^2} = \sqrt{\frac{M^5 T}{L}}\; (\frac{kg m}{A s})&lt;/math&gt;
|-
| Vacuum permeability 
| &lt;math&gt;\frac{a}{2^{11}\pi^5 \Omega^4}&lt;/math&gt;
| &lt;math&gt;\color{red}56\color{black}&lt;/math&gt;
| &lt;math&gt;ux^{\theta} \times uy^7 = \frac{k^{14/5}}{t^{7/{15}}}&lt;/math&gt;  
| &lt;math&gt;\theta - (15 \times 4) \Rightarrow \Omega^{-4}&lt;/math&gt;
| &lt;math&gt;\frac{ux^\theta uy^7}{f(x)^4} = \frac{M^4 T}{L^2}\; (\frac{kg m}{A^2 s^2})&lt;/math&gt;
|}

=== Table of Constants ===
note: &lt;math&gt;\color{red}(u^{15})^n\color{black}&lt;/math&gt; constants have no Omega term.

{| class=&quot;wikitable&quot;
|+Table 9. Dimensioned constants; geometrical vs CODATA 2014
! Constant
! In Planck units
! Geometrical object
! SI calculated (r, v, Ω, α&lt;sup&gt;*&lt;/sup&gt;)
! SI CODATA 2014 &lt;ref&gt;[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)&lt;/ref&gt;
|-
| [[w:Speed of light | Speed of light]]
| V
|  &lt;math&gt;c^* = (2\pi\Omega^2)v,\;u^{17} &lt;/math&gt; 
| ''c&lt;sup&gt;*&lt;/sup&gt;'' = 299 792 458, unit = u&lt;sup&gt;17&lt;/sup&gt; 
| ''c'' =  299 792 458 (exact)
|-
| [[w:Fine structure constant | Fine structure constant]]
|  
| 
| ''α&lt;sup&gt;*&lt;/sup&gt;'' = 137.035 999 139 (mean)
| ''α'' = 137.035 999 139(31)
|-
| [[w:Rydberg constant | Rydberg constant]]
| &lt;math&gt;R^* = (\frac{m_e}{4 \pi L \alpha^2 M})&lt;/math&gt;  
| &lt;math&gt;R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} &lt;/math&gt; 
| ''R&lt;sup&gt;*&lt;/sup&gt;'' = 10 973 731.568 508, unit = u&lt;sup&gt;13&lt;/sup&gt;  
| ''R'' = 10 973 731.568 508(65)
|-
| [[w:Vacuum permeability | Vacuum permeability]]
| &lt;math&gt;\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}&lt;/math&gt; 
| &lt;math&gt;\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}&lt;/math&gt;
| ''μ&lt;sub&gt;0&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 4π/10^7, unit = u&lt;sup&gt;56&lt;/sup&gt; 
| ''μ&lt;sub&gt;0&lt;/sub&gt;'' = 4π/10^7 (exact)
|-
| [[w:Vacuum permittivity | Vacuum permittivity]]
| &lt;math&gt;\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}&lt;/math&gt; 
| &lt;math&gt;\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}&lt;/math&gt;
|  
|
|-
| [[w:Planck constant | Planck constant]]
| &lt;math&gt;h^* = 2 \pi M V L&lt;/math&gt;
| &lt;math&gt;h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}&lt;/math&gt;
| ''h&lt;sup&gt;*&lt;/sup&gt;'' = 6.626 069 134 e-34, unit = u&lt;sup&gt;19&lt;/sup&gt; 
| ''h'' = 6.626 070 040(81) e-34
|-
| [[w:Gravitational constant | Gravitational constant]]
| &lt;math&gt;G^* = \frac{V^2 L}{M}&lt;/math&gt;
| &lt;math&gt;G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{6}&lt;/math&gt; 
| ''G&lt;sup&gt;*&lt;/sup&gt;'' =  6.672 497 192 29 e11, unit = u&lt;sup&gt;6&lt;/sup&gt;
| ''G'' = 6.674 08(31) e-11
|-
| [[w:Elementary charge | Elementary charge]]
| &lt;math&gt;e^* = A T&lt;/math&gt;
| &lt;math&gt;e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{-27}&lt;/math&gt;
| ''e&lt;sup&gt;*&lt;/sup&gt;'' = 1.602 176 511 30 e-19, unit = u&lt;sup&gt;-27&lt;/sup&gt;
| ''e'' = 1.602 176 620 8(98) e-19
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| &lt;math&gt;k_B^* = \frac{2 \pi V M}{A}&lt;/math&gt; 
| &lt;math&gt;k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}&lt;/math&gt; 
| ''k&lt;sub&gt;B&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 1.379 510 147 52 e-23, unit = u&lt;sup&gt;29&lt;/sup&gt;
| ''k&lt;sub&gt;B&lt;/sub&gt;'' = 1.380 648 52(79) e-23
|-
| [[w:Electron mass | Electron mass]]
| 
| &lt;math&gt;m_e^* = \frac{M}{\psi},\; u^{15}&lt;/math&gt; 
| ''m&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 9.109 382 312 56 e-31, unit = u&lt;sup&gt;15&lt;/sup&gt;
| ''m&lt;sub&gt;e&lt;/sub&gt;'' = 9.109 383 56(11) e-31
|-
| [[w:Classical electron radius | Classical electron radius]]
|  
| &lt;math&gt;\lambda_e^* = 2\pi L \psi,\; u^{-13}&lt;/math&gt;  
| ''λ&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 2.426 310 2366 e-12, unit = u&lt;sup&gt;-13&lt;/sup&gt;
| ''λ&lt;sub&gt;e&lt;/sub&gt;'' = 2.426 310 236 7(11) e-12
|-
| [[w:Planck temperature | Planck temperature]]
| &lt;math&gt;T_p^* = \frac{A V}{\pi}&lt;/math&gt;
| &lt;math&gt;T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} &lt;/math&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  1.418 145 219 e32, unit = u&lt;sup&gt;20&lt;/sup&gt;
| ''T&lt;sub&gt;p&lt;/sub&gt;'' = 1.416 784(16) e32
|-
| [[w:Planck mass | Planck mass]]
| M
| &lt;math&gt;m_P^* = (1)\frac{r^4}{v} ,\; \color{red}\color{red}(u^{15})^1\color{black}&lt;/math&gt;
| ''m&lt;sub&gt;P&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' =  .217 672 817 580 e-7, unit = u&lt;sup&gt;15&lt;/sup&gt; 
| ''m&lt;sub&gt;P&lt;/sub&gt;'' = .217 647 0(51) e-7
|-
| [[w:Planck length | Planck length]]
| L
| &lt;math&gt;l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} &lt;/math&gt; 
| ''l&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = .161 603 660 096 e-34, unit = u&lt;sup&gt;-13&lt;/sup&gt;
| ''l&lt;sub&gt;p&lt;/sub&gt;'' = .161 622 9(38) e-34
|-
| [[w:Planck time | Planck time]]
| T
| &lt;math&gt;t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}\color{red}1/(u^{15})^2\color{black} &lt;/math&gt;
| ''t&lt;sub&gt;p&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 5.390 517 866 e-44, unit = u&lt;sup&gt;-30&lt;/sup&gt;
| ''t&lt;sub&gt;p&lt;/sub&gt;'' = 5.391 247(60) e-44
|-
| [[w:Ampere | Ampere]]
| &lt;math&gt;A = \frac{16 V^3}{\alpha P^3}&lt;/math&gt;
| &lt;math&gt;A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 &lt;/math&gt;
| A&lt;sup&gt;*&lt;/sup&gt; = 0.297 221 e25, unit = u&lt;sup&gt;3&lt;/sup&gt;
| ''e/t&lt;sub&gt;p&lt;/sub&gt;'' = 0.297 181 e25 
|-
| [[w:Quantum Hall effect | Von Klitzing constant ]]
| &lt;math&gt;R_K^* = (\frac{h}{e^2})^*&lt;/math&gt;  
|  &lt;math&gt;R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}&lt;/math&gt; 
| ''R&lt;sub&gt;K&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;'' = 25812.807 455 59, unit = u&lt;sup&gt;73&lt;/sup&gt;
| ''R&lt;sub&gt;K&lt;/sub&gt;'' = 25812.807 455 5(59)
|-
| [[w:Gyromagnetic ratio | Gyromagnetic ratio]]
| 
| &lt;math&gt;\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}&lt;/math&gt;  
| ''γ&lt;sub&gt;e&lt;/sub&gt;/2π&lt;sup&gt;*&lt;/sup&gt;'' = 28024.953 55, unit = u&lt;sup&gt;-42&lt;/sup&gt;
| ''γ&lt;sub&gt;e&lt;/sub&gt;/2π'' = 28024.951 64(17) 
|}
 

=== Scalars (general)===

:&lt;math&gt;M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)&lt;/math&gt;

:&lt;math&gt;T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)&lt;/math&gt;

:&lt;math&gt;L = l_p = {2\pi^2\Omega^2}l;\; l = \frac{l_p}{2\pi^2\Omega^2} = .203\;220\;869\;48... 10^{-36},\; u^{-13}\; (m)&lt;/math&gt;

:&lt;math&gt;V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)&lt;/math&gt;

:&lt;math&gt;A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{a})q = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)&lt;/math&gt;


===== MT to LPVA =====

In this example LPVA are derived from MT. The formulas for MT;

:&lt;math&gt;M = (1)k,\; unit = u^{15}&lt;/math&gt;

:&lt;math&gt;T = (\pi) t,\; unit = u^{-30}&lt;/math&gt;

Replacing scalars ''pvlq'' with ''kt''

:&lt;math&gt;P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}&lt;/math&gt;

:&lt;math&gt;V = \frac{2 \pi P^2}{M} = (2 \pi \Omega^2)\; \frac{k^{9/15}}{t^{4/15}},\; unit = u^{9/15*15-4/15*(-30)=17}  &lt;/math&gt;

:&lt;math&gt;L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}&lt;/math&gt;

:&lt;math&gt;A = \frac{2^4 V^3}{a P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{a}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit = 

u^{9/15*(-15)+6/15*30=3}    &lt;/math&gt;


===== PV to MTLA =====

In this example MLTA are derived from PV. The formulas for PV; 

:&lt;math&gt;P = (\Omega)p,\; unit = u^{16}&lt;/math&gt;

:&lt;math&gt;V = (2\pi\Omega^2)v,\; unit = u^{17}&lt;/math&gt;

Replacing scalars ''kltq'' with ''pv''

:&lt;math&gt;M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} &lt;/math&gt;

:&lt;math&gt;T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} &lt;/math&gt;

:&lt;math&gt;L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}&lt;/math&gt;

:&lt;math&gt;A = \frac{2^4 V^3}{a P^3} = (\frac{2^7 \pi^3 \Omega^3}{a})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}&lt;/math&gt;


===Wiki series===
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units MLTPA as geometrical objects
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Physical_constant_(anomaly)]]: Anomalies within the physical constants 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity as a function of atomic orbitals 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model


===Article series===
* [[https://simulationuniverse.org/ simulationuniverse.org]]: Home page
* [[https://simulationuniverse.org/1-Planck-unit-CMB.html 1-Planck-unit-CMB.html]]: Constructs the universe frame from Planck units 
* [[https://simulationuniverse.org/2-Relativity-hypersphere.html 2-Relativity-hypersphere.html]]: Relativity as the mathematics of perspective 
* [[https://simulationuniverse.org/3-Gravitational-orbitals.html 3-Gravitational-orbitals.html]]: Gravity as sum of n-body rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/4-Atomic-orbitals.html 4-Atomic-orbitals.html]]: Atomic orbitals as single rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/5-w_axis.html 5-w_axis.html]]: Imaginary number axis (radiation domain)
* [[https://simulationuniverse.org/6-Physical-constant-anomalies.html 6-Physical-constant-anomalies.html]]: Statistical analysis of physical constant anomalies
* [[https://simulationuniverse.org/7-Monopole-quarks.html 7-Monopole-quarks.html]]: Quarks as monopoles
* [[https://simulationuniverse.org/8-Holographic-universe.html 8-Holographic-universe.html]]: Hypersphere surface as 2-D analogue


=== External links === 
* [https://theprogrammergod.com/ Overview of the Programmer God (mathematical electron model)]


=== References === 
{{Reflist}}


[[Category: Physics]]
[[Category: Philosophy of science]]
__INDEX__</text>
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    <title>Introduction to US History/Immigration &amp; Industrial Revolution</title>
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        <username>Atcovi</username>
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      <text bytes="6535" sha1="qedmvvncpz1cgukyajotljhpcz8mwgl" xml:space="preserve">Americas in the 1850s had to deal a lot with [[U. S. Government/Immigrants|immigration]] and inventions. Most immigrants were from Ireland or Germany. Slavery was also taking new forms of rebellion, including Nat Turner and Harriet Tubman. Some of these new inventions included the cotton gin, Samuel Slater's modification to British textile, John Deer's steel plow, and Lowell Textile Mills.  

An industrial revolution took place because of massive immigration, abundance of land/inventions, and transportation was being made easier (railroads were being developed). {{lecture}}

== Immigration ==
Immigrants found the US to be an attractive destination as there was a lot of land, no mandatory military service, freedom of religion and class society, and lower/no taxes. They took a big risk coming over to America. 30% of immigrants died crossing the Atlantic Ocean. Steamships, including the ''[[w:Titanic|RMS Titanic]]'', were crossing the ocean within 10-12 days. 

As mentioned earlier, a significant number of immigrants were Irish &amp; German. In Ireland, the [[w:Great Famine (Ireland)|1850/1870s Potato Blight]] took place. This resulted in a quarter of the Irish succumbing to starvation due to contaminated crops. As a result of massive Irish immigration, New York City became the world's largest Irish city. As for Germany, crop failure was prevalent and political discontent was an issue. Democracy was sought after by the Germans, but America had already achieved this. Since the Germans were wealther and better educated - their migration was easier.

Some famous immigrants that came to America were Ferdinand Schumacher, Heinrich Steinweg, and Levi Strauss. Schumacher peddled oatmeal in [[US States/Ohio|Ohio]] and created the [[w:Quaker Oaks|Quaker Oaks company]]. Steinweg was a pianist and created [[w:Steinway Pianos|Steinway Pianos]], which was notable for its improved quality and exceptional craft. Levi Strauss designed the &quot;Levi&quot; jeans.

== Industrial Revolution ==

=== Samuel Slater ===
[[File:Samuel Slater industrialist.jpg|thumb|227x227px|&quot;Father of the Factory System&quot;]]
[[w:Samuel Slater|Samuel Slater]] is credited as the &quot;father of the factory system&quot;. The factory system entailed horrid conditions: 12-13hr shifts, 6 days a week with low wages/no labor unions, and unsanitary factories. Child labor was prevalent. He left England illegaly (British mill workers were not allowed to leave the country so that they wouldn't leak the British' way of making cotton/linen fabric) and arrived in Rhode Island in 1791. With the help of fellow inventor Moses Brown, he created the loom and textile mill. 

Slater's '''Factory System''' consisted of 7 boys and 2 girls: malnourished, emotionally abuse, scarred from whips/beatings. Under [[Introduction to US History/1800s America#New Democracy Principle|Jackson's &quot;New Democracy&quot;]], universal suffrage was established so all men can vote. This meant that politicians could advocate for improved working conditions. They were able to rally for a law that made 10 hr workdays the limit. Public education was also demanded and granted. Child labor didn't become an issue until [[President of the United States/Theodore Roosevelt|Theodore Roosevelt]]'s presidency.

==== Lowell Textile Mill ====
The '''Lowell Mills''' were the &quot;state-of-the-art showplace&quot; factory in 1820. This was a place where one could spin cotton into a thread and weave threads into a piece of fabric - in one go. This was also a place where young, single women were encouraged to work. They would endure 10-12 hour workdays and be under monitoring by matrons 24/7.

=== The Southerns' Gem: Cotton Gin ===
{{Info|See the ''[[Introduction to US History/The Impact of the Cotton Gin|essay question]]''}}
[[File:Cotton_gin_EWM_2007.jpg|thumb|A model of the cotton gin]]
The '''Cotton Gin''', made by [[wikipedia:Eli Whitney|Eli Whitney]] in 1793, improved the production of cotton (a single cotton gin could generate nearly 50lbs of cotton) and increased the need for slaves. Eli Whitney applied for a patent on October 29, 1793, but the patient was granted nearly 1 year later on March 14, 1794. The patent wasn't validated until 12 years later in 1807. Not only did the cotton gin lock the slavery system, but it also benefitted the North.

=== Railroads ===
'''Railroads''' were first built in the US in 1828. In 1860, there were over 30k railroad tracks - the majority of it presiding in the North. The railroads prove vital for the North in their transportation of soldiers, which proved fatal for the South in the civil war. 

=== The Reaper ===
[[File:Cyrus_McCormick's_reaper.jpg|left|thumb|Cyrus McCormick's reaper]]
The Reaper, made by Jo Anderson (enslaved African American) and Cyrus McCormick, helped with harvesting grain by separating the grain from the plant stock (1 Reaper = 5 farm hands). Jo Anderson was the real inventor of the Reaper, but Cyrus McCormick improved the reaper and patented the invention. The Reaper was invented in Raphine, VA in McCormick's farm. The patent for the invention was granted on June 21, 1834.

=== John Deere's Steel Plow ===
John Deere, founder of the John Deere company, revolutionized plows by creating plows from steel instead of iron. The plow was self-scouring as sod slipped off the blade. It was light so it could be pulled by only one horse instead of multiple.

=== The Telegraph ===

=== Steamboat ===
[[File:Robert_Fulton_sculpture_IMG_3769.JPG|right|thumb|Robert Fulton, the entrepreneur who improved the steamboat.]]
The Steamboat was improved by entrepreneur Robert Fulton. This improvement changed river transportation since it provided faster transportation from the North to the South.

=== Steam Locomotive ===
The Steam Locomotive provided faster transportation. By 1860, more than 30,000 miles of railroad tracks were made throughout the country. But this tool didn't get a good impression at first, people were skeptical about the train. Going at 18 miles per hour, with bumpy roads, many feared that it could hurt their brain. Some people, by the end of the trail on the Steam Locomotive, would have brain damage. The Steam Locomotive was powered by steam created heated water. Many fires broke out from the soot and dirt when creating the steam. Early train tracks and bridges would be easily broken down, and cause injuries and many casualties.

But soon after the errors were fixed, the steam locomotive was a popular way of getting across the country.
== Navigation ==
{{subpage navbar}}

[[Category:Revolutions]]</text>
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      <text bytes="6544" sha1="ohzzptiayo3ia44z8esw1ixkme2wqtc" xml:space="preserve">Americas in the 1850s had to deal a lot with [[U. S. Government/Immigrants|immigration]] and inventions. Most immigrants were from Ireland or Germany. Slavery was also taking new forms of rebellion, including Nat Turner and Harriet Tubman. Some of these new inventions included the cotton gin, Samuel Slater's modification to British textile, John Deer's steel plow, and Lowell Textile Mills.  

An industrial revolution took place because of massive immigration, abundance of land/inventions, and transportation was being made easier (railroads were being developed). {{lecture}}

== Immigration ==
Immigrants found the US to be an attractive destination as there was a lot of land, no mandatory military service, freedom of religion and class society, and lower/no taxes. They took a big risk coming over to America. 30% of immigrants died crossing the Atlantic Ocean. Steamships, including the ''[[w:Titanic|RMS Titanic]]'', were crossing the ocean within 10-12 days. 

As mentioned earlier, a significant number of immigrants were Irish &amp; German. In Ireland, the [[w:Great Famine (Ireland)|1850/1870s Potato Blight]] took place. This resulted in a quarter of the Irish succumbing to starvation due to contaminated crops. As a result of massive Irish immigration, New York City became the world's largest Irish city. As for Germany, crop failure was prevalent and political discontent was an issue. Democracy was sought after by the Germans, but America had already achieved this. Since the Germans were wealther and better educated - their migration was easier.

Some famous immigrants that came to America were Ferdinand Schumacher, Heinrich Steinweg, and Levi Strauss. Schumacher peddled oatmeal in [[US States/Ohio|Ohio]] and created the [[w:Quaker Oaks|Quaker Oaks company]]. Steinweg was a pianist and created [[w:Steinway Pianos|Steinway Pianos]], which was notable for its improved quality and exceptional craft. Levi Strauss designed the &quot;Levi&quot; jeans.

== Industrial Revolution ==

=== Samuel Slater ===
[[File:Samuel Slater industrialist.jpg|thumb|227x227px|&quot;Father of the Factory System&quot;]]
[[w:Samuel Slater|Samuel Slater]] is credited as the &quot;father of the factory system&quot;. The factory system entailed horrid conditions: 12-13hr shifts, 6 days a week with low wages/no labor unions, and unsanitary factories. Child labor was prevalent. He left England illegaly (British mill workers were not allowed to leave the country so that they wouldn't leak the British' way of making cotton/linen fabric) and arrived in Rhode Island in 1791. With the help of fellow inventor Moses Brown, he created the loom and textile mill. 

Slater's '''Factory System''' consisted of 7 boys and 2 girls: malnourished, emotionally abuse, scarred from whips/beatings. Under [[Introduction to US History/1800s America#New Democracy Principle|Jackson's &quot;New Democracy&quot;]], universal suffrage was established so all men can vote. This meant that politicians could advocate for improved working conditions. They were able to rally for a law that made 10 hr workdays the limit. Public education was also demanded and granted. Child labor didn't become an issue until [[President of the United States/Theodore Roosevelt|Theodore Roosevelt]]'s presidency.

==== Lowell Textile Mill ====
The '''Lowell Mills''' were the &quot;state-of-the-art showplace&quot; factory in 1820. This was a place where one could spin cotton into a thread and weave threads into a piece of fabric - in one go. This was also a place where young, single women were encouraged to work. They would endure 10-12 hour workdays and be under monitoring by matrons 24/7.

=== The Southerns' Gem: Cotton Gin ===
{{Info|See the ''[[Introduction to US History/The Impact of the Cotton Gin|essay question]]''}}
[[File:Cotton_gin_EWM_2007.jpg|thumb|A model of the cotton gin]]
The '''Cotton Gin''', made by [[wikipedia:Eli Whitney|Eli Whitney]] in 1793, improved the production of cotton (a single cotton gin could generate nearly 50lbs of cotton) and increased the need for slaves. Eli Whitney applied for a patent on October 29, 1793, but the patient was granted nearly 1 year later on March 14, 1794. The patent wasn't validated until 12 years later in 1807. Not only did the cotton gin lock the slavery system, but it also benefitted the North.

=== Railroads ===
'''Railroads''' were first built in the US in 1828. In 1860, there were over 30k railroad tracks - the majority of it presiding in the North. The railroads prove vital for the North in their transportation of soldiers, which proved fatal for the South in the civil war. 

=== The Reaper ===
[[File:Cyrus_McCormick's_reaper.jpg|left|thumb|Cyrus McCormick's reaper]]
The Reaper, made by Jo Anderson (enslaved African American) and Cyrus McCormick, helped with harvesting grain by separating the grain from the plant stock (1 Reaper = 5 farm hands). Jo Anderson was the real inventor of the Reaper, but Cyrus McCormick improved the reaper and patented the invention. The Reaper was invented in Raphine, VA in McCormick's farm. The patent for the invention was granted on June 21, 1834.

=== John Deere's Steel Plow ===
John Deere, founder of the John Deere company, revolutionized plows by creating plows from steel instead of iron. The plow was self-scouring as sod slipped off the blade. It was light so it could be pulled by only one horse instead of multiple.

=== The Telegraph ===
{{Stub}}

=== Steamboat ===
[[File:Robert_Fulton_sculpture_IMG_3769.JPG|right|thumb|Robert Fulton, the entrepreneur who improved the steamboat.]]
The Steamboat was improved by entrepreneur Robert Fulton. This improvement changed river transportation since it provided faster transportation from the North to the South.

=== Steam Locomotive ===
The Steam Locomotive provided faster transportation. By 1860, more than 30,000 miles of railroad tracks were made throughout the country. But this tool didn't get a good impression at first, people were skeptical about the train. Going at 18 miles per hour, with bumpy roads, many feared that it could hurt their brain. Some people, by the end of the trail on the Steam Locomotive, would have brain damage. The Steam Locomotive was powered by steam created heated water. Many fires broke out from the soot and dirt when creating the steam. Early train tracks and bridges would be easily broken down, and cause injuries and many casualties.

But soon after the errors were fixed, the steam locomotive was a popular way of getting across the country.
== Navigation ==
{{subpage navbar}}

[[Category:Revolutions]]</text>
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  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2816337</id>
      <parentid>2816301</parentid>
      <timestamp>2026-06-20T14:22:29Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
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      <text bytes="9878" sha1="63xeo6u1ome8pbyz5lopjp3bow9t3er" 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.20260620.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>
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        <username>Alexis Jazz</username>
        <id>791434</id>
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      <comment>Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]])</comment>
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The following data is cached, and was last updated 2026-06-19T11:07:07Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache.
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    <title>WikiJournal Preprints/The unreasonable effectiveness of the cathetus rule in ancient and modern optics</title>
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      <id>2816379</id>
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      <contributor>
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        <id>2838145</id>
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      <text bytes="174130" sha1="awxl99d2t9hwfxdo9rltb8ww7o7au4k" xml:space="preserve">{{Article info
| first = Gavin R.
| last = Putland
| affiliation = Royal Melbourne Institute of Technology, Melbourne, Australia
| orcid = 0000-0003-4757-6341
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| correspondence = [[w:Special:EmailUser/Gavin_R_Putland|Contact form]]
| journal = WikiJournal of Science
| abstract = The &quot;cathetus rule&quot; in optics alleges that the image of an object-point, formed by reflection or refraction at a surface, lies on the perpendicular (&quot;cathetus&quot;) from the object-point to or through the surface. The first known statement of the rule, attributed to Euclid, was for a plane or spherical mirror. The rule was extended to refraction by Ptolemy and to cylindrical and conical mirrors by Ibn al-Haytham, and was upheld by Witelo. But the first valid proofs involving lines of sight other than the cathetus itself were published by Benedetti as late as 1585, for binocular vision, for two special cases: (i) a plane mirror, and (ii) a concave or convex spherical mirror with the two points of reflection (one for each eye) equidistant from the cathetus. Benedetti also gave the first explicit counterexamples to the rule—for a concave or convex spherical mirror with the eyes in the same plane of reflection. Kepler, in 1604, used more general lines of sight than Benedetti, improved on Benedetti's counterexample for the convex spherical mirror, gave the first counterexample for refraction, salvaged the rule for reflection or refraction in a plane or spherical surface subject to appropriate symmetry in the placement of the eyes, offered the first rebuttals of the received rational arguments for the rule, and did all this in a systematic treatise on &quot;the optical part of astronomy&quot;, which so eclipsed Benedetti's book that Kepler was universally credited with the first disproof-and-salvage of the cathetus rule until 2018, when Benedetti's priority was exposed by Goulding.

Kepler notwithstanding, the rule was reaffirmed by Tacquet for plane and spherical mirrors, except for the case in which the rays converge toward a point behind the eye; this became known as the &quot;Barrovian case&quot; because it troubled Barrow, in spite of his modern concept of an image. Barrow demolished the cathetus rule for the tangential image except in the paraxial limit, and Newton salvaged it for the sagittal image. The rule then seems to fade from history.

But the rule is equivalent to the assumption that the image is stigmatic and the cathetus well defined. This narrow assumption is approximately true in the first-order (paraxial, &quot;Gaussian&quot;) analysis of lenses and mirrors; and unacknowledged applications of the ancient rule can indeed be discerned in modern expositions of that subject. Moreover, the validity of the rule for the sagittal image fills a critical gap in meridional ray-tracing through spherical surfaces: by tracing the chief ray from an off-axis object-point, then applying the cathetus rule to the successive surfaces, one can locate successive sagittal image-points on the chief ray (produced rectilinearly through surfaces as necessary), and hence assess astigmatism to leading order, without tracing any rays outside the meridional plane.
| keywords = geometrical optics, Gaussian optics, history of optics, stigmatism, astigmatism, sagittal focus
}}

== Introduction: Undeniable implausibility ==

[[File:Convex mirror.png|thumb|300px|This modern diagram, for locating the image{{mvar| I}}&amp;#8202; of an object-point{{mvar| O}}&amp;#8202; in a convex spherical mirror whose center of curvature is{{mvar| C}}, happens to agree with the ancient cathetus rule. In this case the cathetus is {{mvar|OC&amp;#8202;}} and the point of reflection is{{mvar| V}}.&amp;#8201; According to the rule, the image is at the intersection of the line of sight (through the point of reflection) and the cathetus. (Diagram by &amp;lsquo;Forna&amp;rsquo; at ''Wikimedia Commons''; public domain.)]]

The ''cathetus rule'', as it came to be called, is the ancient optical principle according to which the image of an object-point formed by a reflective or refractive surface lies at the intersection of the line of sight and the ''cathetus'', the latter being the perpendicular let fall from the object-point to the surface. The line of sight and&amp;#10744;or the cathetus may be produced rectilinearly through the surface. In the earliest statements of the rule, but not all statements, the surface is assumed to be plane or spherical. If the premise that the image-point lies on the line of sight is taken as tautological, the rule reduces to the proposition that the image-point lies on the cathetus, but still carries the implication that the line of sight intersects the cathetus.

The rule is easily distilled to an absurdity, especially if we drop the assumption that the surface is plane or spherical. &lt;span id=&quot;active&quot;&gt;Suppose that the image is seen in a part of the surface (which we shall call the ''active'' part) far removed from the cathetus.&lt;/span&gt; If we now deform the surface in a small neighborhood of the cathetus so that the cathetus moves, does the image also move although the object and the observer and the active part of the surface do not? Or if, while preserving the active part, we damage another part of the surface so that there is no longer any cathetus, does the image disappear? For that matter, does the image disappear—even in a plane or spherical surface—if we merely cover the point on the surface where the cathetus falls?

== History ==

=== Euclid ===

In the oldest surviving source of the cathetus rule, namely the ''[[w:Catoptrics|Catoptrics]]'' traditionally attributed to [[w:Euclid|Euclid]], the last-mentioned absurdity seems to be not only tolerated as an implication, but relied upon as a premise, and even stated among the postulates at the outset: the 4th and 5th postulates, as paraphrased by [[w:A. Mark Smith|A.&amp;#8239;Mark Smith]], state that in plane, convex spherical, and concave spherical mirrors, &quot;if a perpendicular (the so-called cathetus) is dropped from an object to the mirror's surface, and if the point at which it meets that surface is covered, the object will no longer be seen.&quot;&lt;ref&gt;[[#smith-2017|Smith, 2017]], p.&amp;#8239;56. For the original Greek and the Latin translation by Jean Pena, see [[#euclid-pena-1557|Euclid/Pena, 1557]], p.&amp;#8239;35 in the Greek version, &amp;amp; p.&amp;#8239;45 in the Latin version. Smith evidently follows a different edition in numbering the offending postulates as 4 and 5; though I have small Greek and less Latin, I notice that Pena's edition divides the corresponding postulates into nos.&amp;#8239;4,&amp;#8239;5,&amp;#8239;and 6, referring respectively to plane, convex spherical, and concave spherical mirrors.&lt;/ref&gt; Euclid cites these postulates, together with the premise that the image lies on the line of sight (Postulate 2), to prove the cathetus rule for plane mirrors (Proposition 16), convex spherical mirrors (Proposition 17), and concave spherical mirrors (Proposition 18).&lt;ref&gt;See [[#euclid-pena-1557|Euclid/Pena, 1557]], p.&amp;#8239;42 in the Greek &amp;amp; pp.&amp;#8239;55–6 in the Latin.&lt;/ref&gt;

&lt;span id=&quot;takahashi-defense&quot;&gt;[[w:Ken'ichi Takahashi|Ken'ichi Takahashi]] has suggested, in Euclid's defense, that the 4th and 5th postulates refer correctly to the case in which the observer looks along the cathetus, so that the line of sight is blocked by the object&lt;/span&gt;,&lt;ref&gt;[[#takahashi-92|Takahashi, 1992]], pp.&amp;#8239;20–26, cited by [[#smith-2017|Smith, 2017]], pp.&amp;#8239;59–61, and by [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;500–501.&lt;/ref&gt; or, I should add, by the observer's head, if it is between the object and the mirror. Under that interpretation, the cathetus rule seems to be based on the reasonable premise that the image-point lies at the intersection of ''two'' lines of sight. But that does not explain why the cathetus (if it exists) must be one of them, or why all choices of the other should intersect the cathetus at the same point (if at all), or how we can speak of &quot;the&quot; image if they do not. Neither does any &quot;two lines of sight&quot; argument appear in subsequent ancient and medieval efforts to defend the rule (as we shall see). Nevertheless the rule is upheld as often as it is mentioned, for both reflection and refraction, by all optical writers until [[w:Giambattista Benedetti|Benedetti]] (1585),&lt;ref&gt;[[#goulding-18|Goulding, 2018]].&lt;/ref&gt; and by all better-known ones until Kepler.&lt;ref&gt;[[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;26–7.&lt;/ref&gt;

[[w:Johannes Kepler|Johannes Kepler]], in the third chapter of his ''Paralipomena'' (1604), initially interprets Euclid's premise in the more literal, absurd manner, and duly dismisses it. Supposing that ''C''&amp;#8202; is the foot of the cathetus from the object-point ''A'',&amp;#8239; Kepler says of Euclid:
&lt;blockquote&gt;That the place of the image of the object ''A'' is on ''AC''&amp;#8202; he proves thus: &quot;For,&quot; he says, &quot;when the position ''C''&amp;#8202; of the mirror is taken, upon which the perpendicular falls, the visible object ''A'' is no longer seen.&quot; If by &quot;taken&quot; you understand &quot;occupied&quot; (that is, that the position ''C''&amp;#8202; is covered), the axiom is false…&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;73. In the quotation, which appears in italics in the original edition ([[#kepler-1604|Kepler, 1604]], p.&amp;#8239;56), Kepler may be translating from Greek, or paraphrasing, rather than quoting from Latin; ''cf''. [[#euclid-pena-1557|Euclid/Pena, 1557]], p.&amp;#8239;42 in the Greek &amp;amp; p.&amp;#8239;55 in the Latin.&lt;/ref&gt;
&lt;/blockquote&gt;
Kepler offers Euclid a lifeline but cannot save him:
&lt;blockquote&gt;Let us now grant that Euclid's axiom is to be understood differently, so as to state that if the observer were situated at ''A''&amp;#8201; and ''C''&amp;#8202; were covered, then ''A'' would not be seen. Then the axiom is perfectly true, but the conclusion does not follow from it, except for perpendicular viewing. The argument does not carry over from a perpendicular to an oblique observer.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;74.&lt;/ref&gt;
&lt;/blockquote&gt;
Kepler's lifeline is not as general as Takahashi's; but even if it were, the argument would still &quot;not carry over&quot; to an oblique viewer, in as much as it would not explain the distance of the perceived image along a ''single'' line of sight, or why that line of sight should intersect the cathetus.

In Euclid's Postulate 4 and Proposition 16, the Greek ''káthetos'' is rendered in Latin as ''perpendicularis''&amp;#8202; by at least three translators,&lt;ref&gt;[[#euclid-pena-1557|Euclid/Pena, 1557]], pp.&amp;#8239;45 &amp;amp; 55 in the Latin; [[#euclid-dasypodius-1557|Euclid/Dasypodius, 1557]] (unnumbered pages); [[#euclid-heiberg-1895|Euclid/Heiberg, 1895]], pp.&amp;#8239;286–7,&amp;#8239;312–13.&lt;/ref&gt; whereas Postulate 5 and Propositions 17 and 18 refer to the cathetus not by any name, but as the line drawn to the center of the sphere.

=== Ptolemy ===

When interpreting the authorities on geometrical optics before 1000{{midsize|&amp;#8239;CE}}, we must remember that they believed in visual rays emitted by the eye, so that the &quot;incident&quot; ray is from the eye, not from the object-point; the &quot;cathetus of incidence&quot; (if it is mentioned) is therefore the perpendicular from the ''eye'' to the surface, while the usual &quot;cathetus&quot; (the perpendicular from the ''object-point'' to the surface) may be called the cathetus of reflection or refraction. So it is with [[w:Ptolemy|Ptolemy]]'s ''Optics'', written some years after his ''[[w:Almagest|Almagest]]'', but known to us only through a 12th-century Latin translation of a now lost, incomplete Arabic translation.&lt;ref&gt;[[#smith-1996|Smith, 1996]], pp.&amp;#8239;1–8; [[#lindberg-81|Lindberg, 1981]], p.&amp;#8239;211.&lt;/ref&gt; (Even the Latin version was not available to Kepler,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;84, n.&amp;#8239;34; [[#lohne-59|Lohne, 1959]], pp.&amp;#8239;117-18.&lt;/ref&gt; and not printed until 1885.&lt;ref&gt;[[#ptolemy-govi-1885|Ptolemy/Govi, 1885]].&lt;/ref&gt;)

Ptolemy affirms the cathetus rule for reflection in a plane or spherical mirror, on the empirical ground that a thin rod standing perpendicularly on the reflecting surface appears aligned with its reflection&lt;ref&gt;[[#smith-2017|Smith, 2017]], p.&amp;#8239;93.&lt;/ref&gt; when &quot;properly viewed outside the mirror.&quot;&lt;ref&gt;[[#smith-1996|Smith, 1996]], pp.&amp;#8239;131–2.&lt;/ref&gt; That premise is certainly true if the rod is viewed with one eye, due to the axial symmetry about the rod (the cathetus), implying a bilateral symmetry (&quot;mirror symmetry&quot;) about the plane of the eye and the rod. But it proves only that the image is in that plane—not that it is necessarily collinear with the rod. Moving the eye around the rod does not prove anything more, because the said plane moves with the eye, so that the image, if not collinear with the rod, moves with the plane.

Ptolemy then notes that the perpendicular to the surface at the point of reflection is in the plane of the line of sight and the cathetus,&lt;ref&gt;[[#smith-1996|Smith, 1996]], p.&amp;#8239;132.&lt;/ref&gt; which is indeed the case if we retain the symmetry. Thus he makes the cathetus rule the ''premise'' of an aspect of the law of reflection—an aspect that seems to have escaped his predecessors&lt;ref&gt;[[#smith-1996|Smith, 1996]], p.&amp;#8239;36.&lt;/ref&gt;— namely that the incident and reflected rays and the normal at the point of reflection are coplanar!&lt;ref&gt;''Cf''. [[#goulding-18|Goulding, 2018]], p.&amp;#8239;502.&lt;/ref&gt;

&lt;span id=&quot;floating-coin&quot;&gt;Later in his treatise, Ptolemy makes the corresponding aspect of the law of ''refraction'' dependent on the cathetus rule. As evidence for the latter, he cites the already old &quot;floating coin&quot; experiment,&lt;/span&gt; in which a coin lying on the bottom of a tub and hidden by the rim is seemingly raised into view by filling the tub with water.&lt;ref&gt;[[#smith-1996|Smith, 1996]], pp.&amp;#8239;230–31.&lt;/ref&gt; He does not explain why the image should be raised precisely ''vertically'', as the cathetus rule requires—and as will ''appear'' to be confirmed in observations that tacitly exploit the axial symmetry. And although the cited experiment concerns a ''plane'' refracting surface, Ptolemy goes on to apply the rule to spherical refracting surfaces without further justification.&lt;ref&gt;[[#smith-1996|Smith, 1996]], pp.&amp;#8239;252–3.&lt;/ref&gt;

In addition to these flawed empirical demonstrations of the rule, Ptolemy attempts a rational explanation, saying that the location of the image must be unique, and that &quot;to any point on a given object there is one and only one cathetus, whereas any other line, being oblique with respect to this cathetus, is subject to numerous variations.&quot;&lt;ref&gt;Translated by Smith ([[#smith-1996|1996]], p.&amp;#8239;138); cited by Goulding ([[#goulding-18|2018]], p.&amp;#8239;503).&lt;/ref&gt; There are at least two weaknesses in this argument. First, some qualification must be imposed on the image-point in order to ensure uniqueness; Ptolemy himself shows that for given positions of the object-point and the eye, a concave mirror can give multiple points of reflection, which, according to the cathetus rule, will give multiple image-points on a common cathetus.&lt;ref&gt;[[#smith-2017|Smith, 2017]], p.&amp;#8239;102; [[#goulding-18|Goulding, 2018]], p.&amp;#8239;505''n''.&lt;/ref&gt; Second, and more seriously, even if the image-point and the cathetus are both unique, that does not prove any other connection between the two!

In the Latin text of Ptolemy's ''Optics'', which is already a translation of a translation, the cathetus is again called the ''perpendicularis''.&lt;ref&gt;[[#smith-1996|Smith, 1996]], pp.&amp;#8239;287–8, 296. The word ''cathetus'' and the expressions ''cathetus of incidence'' and ''cathetus of reflection'' appear in Smith's English translation, and these terms together with ''cathetus of refraction'' appear in his annotations.&lt;/ref&gt;

&lt;br /&gt;

In antiquity, the cathetus rule was found effective in spite of its lack of foundation, and not only for establishing the coplanarity laws of reflection and refraction. Its effectiveness for ''reflection''&amp;#8202; is accidentally emphasized by one medieval author who seems unfamiliar with the rule: the Syrian Christian polymath [[w:Qusta ibn Luqa|Qusṭā ibn Lūqā]] (820?–912?{{midsize|&amp;#8239;CE}}). In only one case—that of a plane mirror—does he specify the location of a reflected image. To explain why (e.g.) the image in a convex mirror is diminished,&lt;ref&gt;[[#smith-2017|Smith, 2017]], pp.&amp;#8239;170–71.&lt;/ref&gt; Ibn&amp;#8239;Lūqā compares the apparent extent of the image ''on the reflecting surface'' with that given by a plane mirror—whereas Euclid&lt;ref&gt;[[#smith-2017|Smith, 2017]], p.&amp;#8239;61.&lt;/ref&gt; and Ptolemy,&lt;ref&gt;[[#smith-1996|Smith, 1996]], pp.&amp;#8239;165–9.&lt;/ref&gt; aided by the cathetus rule, have correctly deduced not only that the image is diminished, but also that it is closer to the reflective surface than the object is, and that convex mirrors make the world look convex.

By the end of the 10th century, however, Ptolemy's ''Optics'' has been translated into Arabic,&lt;ref&gt;[[#smith-1996|Smith, 1996]], p.&amp;#8239;6.&lt;/ref&gt; ready to be studied—and surpassed—by &quot;the most significant figure in the history of optics between antiquity and the seventeenth century.&quot;&lt;ref&gt;[[#lindberg-81|Lindberg, 1981]], p.&amp;#8239;58.&lt;/ref&gt;

=== Alhacen ===

Abū ‘Alī al-Ḥasan (&quot;Alhacen&quot;) ibn al-Ḥasan [[w:Ibn al-Haytham|ibn al-Haytham]]{{efn|The original Latinization of his name was ''Alhacen'', not the more familiar ''Alhazen'' ([[#lindberg-81|Lindberg, 1981]], pp.&amp;#8239;209–10; [[#smith-2017|Smith, 2017]], p.&amp;#8239;1).}} wrote his ''Book of Optics'' circa 1030{{midsize|&amp;#8239;CE}}.&lt;ref&gt;[[#smith-2017|Smith, 2017]], p.&amp;#8239;182.&lt;/ref&gt; For Alhacen, the eye is not an emitter of visual rays, but a receiver of light rays.&lt;ref&gt;[[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;17–18; [[#smith-2017|Smith, 2017]], pp.&amp;#8239;184–6.&lt;/ref&gt;{{efn|Although Alhacen's theory of vision was not the first ''intromission'' theory, it was apparently the first such theory to incorporate the premise (first stated explicitly by [[w:al-Kindi|al-Kindī]] in the 9th century) that each visible spot on a luminous or illuminated body sends out ''light'', and consequently the first such theory that could be reconciled with a geometrical science of optics ([[#lindberg-81|Lindberg, 1981]], pp.&amp;#8239;30–31,&amp;#8239;58–60).}} Hence, in reflection or refraction, the &quot;incident&quot; ray is not from the eye, but from the object-point, and the &quot;perpendicular of incidence&quot; is dropped from the object-point, while the &quot;line of sight&quot; now coincides with the &quot;line of reflection&quot; or the &quot;line of refraction&quot;. This reversal of direction does not affect the geometry and therefore does not of itself furnish any new arguments for the cathetus rule, although Alhacen offers many—some empirical and some rational, for both reflection&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;385–97.&lt;/ref&gt; and refraction&lt;ref&gt;[[#smith-2010|Smith, 2010]], pp.&amp;#8239;275–82.&lt;/ref&gt;&amp;#8202;—none of which is an exemplar of the rigor for which he is otherwise renowned.

In the ''empirical'' category, for a plane mirror,&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;385–7.&lt;/ref&gt; Alhacen recommends putting marks on Ptolemy's rod (but does not name Ptolemy here). Then he tries a cone instead of a rod, and invites us to imagine such a cone extended to the mirror from every point on the object. He notes that the same observations hold for convex spherical mirrors.&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;387–8.&lt;/ref&gt; Conceding that they do ''not'' generally hold for a convex ''cylindrical'' mirror, because &quot;what is straight does not appear straight&quot;, Alhacen claims that the cathetus rule is still verified for a ''single point'' on the object seen in such a mirror.&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;388.&lt;/ref&gt; It seems to escape his notice that if the image of a point on a thin rod standing perpendicularly on the mirror does not align with the rod, then the line of reflection, when produced through the mirror, does not intersect the cathetus at all. Obviously, by symmetry, the image will appear to align with the rod if the plane of the eye and the rod contains the axis of the cylinder or is perpendicular thereto; and his experiments confirm these cases.&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;388–9.&lt;/ref&gt; In intermediate cases, if the image of the tip of the rod is to fall on the cathetus, the line of sight and therefore the point of reflection must be in the plane of the eye and the cathetus, so that the point of reflection must be on the elliptical section of that cylinder by the plane—which is precisely what Alhacen claims,&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;389–91 (par.&amp;#8239;2.15–18) and note 12 (p.&amp;#8239;489), referring to figure 5.2 on p.&amp;#8239;216 (other volume).&lt;/ref&gt; without checking the requirement that the normal to the cylinder at this point is in the same plane (that of the incident and reflected rays), as he stipulates in his statement of the law of reflection.&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;300.&lt;/ref&gt;

After briefly claiming that the same procedure can be applied to convex conical mirrors, with the same results (!), Alhacen turns to concave spherical mirrors.&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;391–4.&lt;/ref&gt; Fashion a right circular cone whose slant height is equal to the radius of curvature of the mirror, mark a &quot;line of longitude&quot; (generating line) on the cone, and mount the cone on the mirror, so that the apex of the cone is at the center of curvature of the mirror; then, he says, the cone and the line of longitude will appear to extend into the mirror. Next, having placed the apex at the center of curvature, mount a thin rod on the mirror so that its tip is between the apex and the mirror while the image of the tip is in front of the mirror; then the image will be nearer to the eye (note the singular) than the apex is, and you will be able to bring the tip, the apex, and the image into a single line of sight. Finally he claims that the cathetus rule holds for concave cylindrical and conical mirrors, by the same flawed reasoning as for their convex counterparts.

In the account of the concave spherical mirror, modern readers will recognize the apparent continuation of the cone into the mirror as the virtual image of an object inside focus, and will recognize the image of the tip of the rod as the real image of an object-point between the focus and the center of curvature. Otherwise the above observations of Alhacen, in so far as they are correct, are trivial consequences of the axial symmetry of the surface about the cathetus or catheti; and in only one case—that in which we look along the cathetus, through the image of the rod-tip to the tip itself—does he establish that the image is on the cathetus and not merely in the plane of the cathetus and the eye.

[[File:Alhacen-disk-experiment.jpg|thumb|256px|Kitchen-bench reconstruction of Alhacen's first experiment attempting to prove the cathetus rule for refraction. A vertical diameter and a sloping diameter are drawn on the base of a coffee mug. The vertical diameter appears to continue vertically into the water, showing that the ''image'' of the point of intersection is in the plane of the viewing position and the cathetus (vertical diameter). Alhacen would claim that the image-point is not only in this plane, but ''on'' the cathetus. (Photo by the author; public domain.)]]

For refraction, Alhacen rightly cites the [[#floating-coin|floating-coin experiment]] as proof that the image is displaced from the object.&lt;ref&gt;[[#smith-2010|Smith, 2010]], pp.&amp;#8239;274–5.&lt;/ref&gt; He then asserts the cathetus rule, and claims to prove it by a variant experiment in which a vertical diameter and a sloping diameter are marked on a vertical disk, which is immersed in water up to a point above the intersection (center of the disk), with the marked surface facing the eye (note the singular), which is best placed just above the water level. The vertical line then appears to continue vertically into the water, so that the point of intersection (the object-point) appears to lie on the continuation (the cathetus), while the sloping line appears to be kinked at the surface. Alhacen further recommends rotating the disk so as to interchange the roles of the two marked diameters.&lt;ref&gt;[[#smith-2010|Smith, 2010]], pp.&amp;#8239;275–7.&lt;/ref&gt; But, whichever diameter is the cathetus, he again fails to explain why the image is on the cathetus and not merely in the plane of the cathetus and the eye.

This defect is not repaired by the next experiment, using the same disk but no water, which is intended to interchange the places of the rare and dense media.&lt;ref&gt;[[#smith-2010|Smith, 2010]], pp.&amp;#8239;277–80.&lt;/ref&gt; A rectangular glass block, with its top and bottom faces horizontal, is affixed to the disk near the top, covering a portion of each marked diameter. The observer's eyes are positioned so that one eye is close to the top face of the block and sees both diameters through the block, while the other eye sees the intersection without refraction (bypassing the block). Then the former eye perceives the entire vertical diameter (the cathetus) as vertical and aligned with the portion seen by the other eye without refraction, although the two eyes see the intersection at different points on the cathetus. Thus the image of the intersection, as seen by the former eye, appears to be on the cathetus. But again this appearance follows from the weaker condition that each eye perceives the vertical diameter (or the relevant part thereof) to be in the ''plane'' of that eye and the vertical diameter: as the two planes intersect on the vertical diameter, that diameter must appear in its true alignment, even if the eyes disagree on the positions of its constituent points (only one of which—the intersection—looks different from the others).

In the ''rational'' category, for reflection, Alhacen sets out to explain &quot;why visible objects are perceived through reflection where the image is located and why the image lies on the normal from the visible object to the surface of the mirror.&quot;&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;394.&lt;/ref&gt; On the latter question, he first says that we judge the distance of an image by comparing its angular size with its absolute size.&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;395.&lt;/ref&gt; For the purpose of establishing the cathetus rule, by which we propose to locate points on images and thence determine absolute sizes of images, this is a circular argument.

For plane mirrors, says Alhacen, &quot;since the image does not appear on the surface of the mirror but behind it, it is more appropriate and reasonable for it to appear upon rather than outside the normal.&quot;&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;395; ''cf''. [[#goulding-18|Goulding, 2018]], p.&amp;#8239;504.&lt;/ref&gt; Taking that as a ''premise'', he correctly locates the image. He adds that if the image were beyond or in front of the cathetus, then, since the image lies on the line of reflection, it would be further from or nearer to the eye and would therefore subtend a smaller or larger angle.&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;396.&lt;/ref&gt; But in fact, according to the law of reflection, it would subtend the ''same'' angle because the line of reflection from each point on the object would be unchanged. Kepler raises another objection: Alhacen &quot;says that when an image is perceived on the perpendicular, it has the proper magnitude belonging to the thing itself.&quot; But this magnitude, as Kepler notes, cannot be a necessary condition for the correct location of the image, because it does not hold for curved mirrors.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;75.&lt;/ref&gt;

For a convex mirror, Alhacen argues verbosely but validly that the image of the center of the eye (note the singular) must be on the cathetus due to symmetry. But then he extends the argument to the image of any other point on the eye, although the symmetry is broken in that the image is no longer ''seen'' along the cathetus; and he briefly claims that the same logic applies to a concave spherical mirror and to a concave or convex conical mirror,&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;396–7.&lt;/ref&gt; although in the conical case, even the surface is not axially symmetrical about the cathetus.

Just before the claim on concave and conical mirrors, Alhacen says in support of the cathetus rule:
&lt;blockquote id=&quot;alhacen-obj-img&quot;&gt;The state of natural things is in accordance with the situation of their principles, and the principles of natural things are hidden.&lt;ref&gt;Quoted in translation by Goulding ([[#goulding-18|2018]], p.&amp;#8239;505); ''cf''. [[#smith-2006|Smith, 2006]], p.&amp;#8239;397.&lt;/ref&gt;
&lt;/blockquote&gt;
&quot;By these words he says two things,&quot; says Kepler. &quot;First, he repeats the very thing that was proposed to prove (for they say nothing different), and second, he says by way of appending the cause, that it is hidden. But this is not demonstrating.&quot;&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;74–5.&lt;/ref&gt; And just ''after'' the claim on concave and conical mirrors, Alhacen continues:
&lt;blockquote&gt;And the place of the image will universally be on the perpendicular in any mirror, because there is no place outside the perpendicular in which the form maintains a likeness and identity of position.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], p.&amp;#8239;505; ''cf''. [[#smith-2006|Smith, 2006]], p.&amp;#8239;397.&lt;/ref&gt;
&lt;/blockquote&gt;
Thus he seems to argue from the location of the thing seen to the location of the image; this mode of reasoning will reappear later.

Broadening the attack, Kepler adds: &quot;''But this fact further strongly confutes the Optical writers'', that they do not give the same cause of this matter in reflection as in refraction.&quot;&lt;ref&gt;Italics in the Latin ([[#kepler-1604|Kepler, 1604]], p.&amp;#8239;58), not quite matching [[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;75.&lt;/ref&gt; Indeed, in support of the cathetus rule for refraction, Alhacen apparently reasons that the motion of the light ray in the medium containing the object-point can be resolved into a component in the direction of the cathetus, and a component perpendicular thereto.&lt;ref&gt;[[#smith-2010|Smith, 2010]], pp.&amp;#8239;280–82.&lt;/ref&gt; An obvious weakness in that argument, if we credit it with any relevance at all (which Kepler does not), is that we can choose the former direction differently and still perform the resolution. Kepler also argues, somewhat cryptically, that refraction further weakens Alhacen's connection between image size and correct image location, in that the size-distance relation for refraction is different from that for reflection.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;76.&lt;/ref&gt;

On three pillars—the cathetus rule, the correct law of reflection, and an incomplete law of refraction—Alhacen builds a comprehensive and largely correct theory of image location, magnification, and distortion in seven types of mirrors,&lt;ref&gt;[[#smith-2017|Smith, 2017]], pp.&amp;#8239;204–5.&lt;/ref&gt; image location in plane and spherical refracting surfaces,&lt;ref&gt;[[#smith-2010|Smith, 2010]], chap.&amp;#8239;5, par.&amp;#8239;25–90 (p.&amp;#8239;282ff).&lt;/ref&gt; and magnification by spherical refracting surfaces.&lt;ref&gt;[[#smith-2010|Smith, 2010]], chap.&amp;#8239;7.&lt;/ref&gt; Taking the first pillar to imply that an object-point is perceived to lie on the cathetus from that point to the outer refracting surface of the ''eye'', he even offers an explanation why the eye perceives the direction of the object-point although light from that point strikes all points of the eye.&lt;ref&gt;[[#smith-2010|Smith, 2010]], pp.&amp;#8239;303–4, par.&amp;#8239;6.22–3. ''Cf''. [[#lindberg-81|Lindberg, 1981]], pp.&amp;#8239;76–78. Remarkably, this is the only context in which Lindberg (1981) mentions the cathetus rule (which he states but does not name). More remarkably, he says that in general the rule &quot;makes perfect sense, for it requires simply that the eye be unaware of the break in the ray… and therefore that it project the image backward along the incident ray&quot; (p.&amp;#8239;76). Apart from misidentifying the ray (an obvious and temporary slip), this explanation fails to explain ''how far'' the image should be projected back.&lt;/ref&gt;{{efn|But Alhacen does not question the ancient, erroneous doctrine that the glacial humor (lens) is the sensitive part of the eye ([[#smith-2001|Smith, 2001]], p.&amp;#8239;417, par.&amp;#8239;2.1). Nor does he deduce (as he would in any other context) that the image-point lies behind the center of the eye (as it does), because that would give an inverted image (as it does), which apparently would imply that we see upside-down! Instead, he concludes that there must be a diverging refraction at the back surface of the glacial humor, so that the cathetal rays from the various object-points do not cross each other ([[#smith-2001|Smith, 2001]], pp.&amp;#8239;419–20). ''Cf''. [[#lindberg-81|Lindberg, 1981]], pp.&amp;#8239;76–78,&amp;#8239;80–81.}} Yet neither he nor anyone before him has offered a firm foundation for that first pillar.

[[File:Plane-mirror-opt-cropped.svg|thumb|300px|Location of the image{{mvar| P&amp;prime;}} of an object-point{{mvar| P}}&amp;#8202; in a plane mirror{{mvar| B}}.&amp;#8201; In this special case the cathetus rule follows simply and rigorously from the law of reflection (although Ptolemy and Alhacen still cite the cathetus rule independently). (Original diagram by &amp;lsquo;MikeRun&amp;rsquo; at ''Wikimedia Commons''.)]]

&lt;span id=&quot;ingredients&quot;&gt;For the case of reflection in a plane mirror, however, the ''ingredients'' of a valid proof of the cathetus rule have been unwittingly served up by Ptolemy and Alhacen.&lt;/span&gt; From the law of reflection ''and the cathetus rule'', Ptolemy proves that the image-point is as far behind the mirror as the object-point is in front.&lt;ref&gt;[[#smith-1996|Smith, 1996]], pp.&amp;#8239;155–6 (Theorem {{serif|III}}.5), summarized in [[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;13–14.&lt;/ref&gt; If the cathetus rule is not assumed ''a priori'', the same geometric argument simply shows that the reflected line of sight to the object-point, when produced from the eye through the mirror, intersects the cathetus as far behind the mirror as the object-point is in front (provided that the line of sight intersects the cathetus at all, as is obvious from the symmetry). By the generality of this line of sight, all such lines of sight intersect the cathetus at the same point, and therefore intersect each other at a common point—a ''[[w:Stigmatism|stigmatic]]'' image—which is ''on the cathetus''. But Ptolemy does not package the argument that way. Nor does Alhacen, who again shows that the line of sight intersects the cathetus as far behind the mirror as the image-point is in front.&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;399 (pars.&amp;#8239;2.47–8 in Prop.&amp;#8239;4).&lt;/ref&gt;

=== The three friars ===

In the West, as [[w:David C. Lindberg|David C. Lindberg]] explains,
&lt;blockquote&gt;the character of the twelfth-century revival of learning was dramatically transformed by a flood of translations from both Greek and Arabic; what was at first chiefly an intensification of interest in ancient Latin sources became a quest for new knowledge, previously unavailable in the West.&amp;#8239;… In optics, …&amp;#8239;it was not until the middle of the thirteenth century that the full corpus of Greek and Arabic works on the subject was at hand in the major European centers of learning, able to shape (and indeed revolutionize) the thought of Western scholars.&lt;ref&gt;[[#lindberg-81|Lindberg, 1981]], pp.&amp;#8239;102–3.&lt;/ref&gt;
&lt;/blockquote&gt;
Foremost in the &quot;corpus&quot; is Alhacen's ''Book of Optics'', translated into Latin circa 1200 as ''De&amp;nbsp;Aspectibus''. This is the main source, but not the only source, for the three leading Western &quot;perspectivist&quot;&lt;ref&gt;The term was coined by Lindberg ([[#lindberg-81|1981]], p.&amp;#8239;251, n.&amp;#8239;1) from the late medieval Latin equivalent.&lt;/ref&gt; works, namely
* [[w:Roger Bacon|Roger Bacon]]'s ''Perspectiva'', written circa 1263, and dispatched to the papal court as part 5 of his ''Opus Majus'' in 1267 or 1268,
* [[w:Vitello|Witelo]]'s ''Perspectiva'', written at the papal court, probably in the first half of the 1270s, and
* [[w:John Peckham|John Pecham]]'s ''Perspectiva Communis'', probably written at the papal court in the late 1270s, just before the author's appointment as Archbishop of Canterbury.&lt;ref&gt;[[#lindberg-71|Lindberg, 1971]], pp.&amp;#8239;68–9,&amp;#8239;71 (on Bacon), pp.&amp;#8239;72–3 (on Witelo), pp.&amp;#8239;82–3 (on Pecham).&lt;/ref&gt;

Bacon, according to Lindberg, is the first Western optical writer to cite Ptolemy's ''Optics'', and only the third to use Alhacen's ''De&amp;nbsp;Aspectibus''.&lt;ref&gt;[[#lindberg-81|Lindberg, 1981]], p.&amp;#8239;253, n.&amp;#8239;28.&lt;/ref&gt; He also draws on Euclid's ''Catoptrics'' in a circular attempt to establish the cathetus rule, which he then applies in selected cases.&lt;ref&gt;[[#smith-2017|Smith, 2017]], pp.&amp;#8239;267–8.&lt;/ref&gt; Thus he becomes, as far as I have noticed in this brief inquiry, the first author to use the Latin term ''cathetus'' in the optical sense—mostly in the phrase ''cum catheto'' (&quot;with the cathetus&quot;).&lt;ref&gt;[[#bacon-combach-1614|Bacon/Combach, 1614]], is digitally searchable.&lt;/ref&gt;

Witelo is clearly familiar with Bacon's work, presumably through the patronage of the papal confessor (and prolific translator of ancient Greek treatises), [[w:William of Moerbeke|William of Moerbeke]].&lt;ref&gt;[[#lindberg-71|Lindberg, 1971]], pp.&amp;#8239;72–5.&lt;/ref&gt; But, whereas Bacon summarizes ''De&amp;nbsp;Aspectibus'', Witelo expands on it, incorporating material from Euclid, [[w:Hero of Alexandria|Hero of Alexandria]], Ptolemy, al-Kindī, Alhacen's treatise on parabolic burning mirrors, and [[w:Ibn Mu'adh al-Jayyani|Ibn&amp;#8239;Mu‘ādh]]'s essay on twilight, rearranging the content with a mathematical introduction and a consistent theorem-and-proof format—suitable for a textbook or reference—and adding a theological prolog for a Roman Catholic readership.&lt;ref&gt;[[#smith-2017|Smith, 2017]], pp.&amp;#8239;273–5; [[#unguru-72|Unguru, 1972]].&lt;/ref&gt; And whereas the Latin text of Alhacen's ''De&amp;nbsp;Aspectibus'' does not seem to contain the word ''cathetus'' or any inflected form thereof (although ''perpendicularis'' and ''perpendiculari'' are ubiquitous), Witelo's ''Perspectiva'' uses that word in some form more than 150 times, including at least 19 occurrences of the phrase ''cum catheto''.&lt;ref&gt;[[#risner-1572|Risner, 1572]], is digitally searchable.&lt;/ref&gt;

Pecham also is clearly familiar with Bacon's work, probably through personal acquaintance, both men having joined the [[w:Franciscans|Franciscan]] order at Oxford in the 1250s and resided at the Franciscan convent in Paris in the 1260s.&lt;ref&gt;[[#lindberg-71|Lindberg, 1971]], pp.&amp;#8239;75–7.&lt;/ref&gt;{{efn|Witelo, according to various modern sources, was also a friar; but I have not been able to establish the order to which he belonged, or whether this is known. Moerbeke was a [[w:Dominican Order|Dominican]].}} Pecham, like Bacon, summarizes Alhacen, but follows him more closely,&lt;ref&gt;[[#smith-2017|Smith, 2017]], pp.&amp;#8239;273–5.&lt;/ref&gt; and again uses the expressions ''cathetus'' and ''cum catheto''.&lt;ref&gt;[[#pecham-gaurico-1504|Pecham/Gaurico, 1504]], and [[#pecham-hartmann-1542|Pecham/Hartmann, 1542]], are digitally searchable.&lt;/ref&gt;{{efn|Lindberg ([[#lindberg-71|1971]], pp.&amp;#8239;66,&amp;#8239;77–83) offers evidence that Pecham was also indebted to Witelo through Moerbeke, but notes that the citations of Witelo in the ''Perspectiva Communis'' are spurious, having been introduced by [[w:Georg Hartmann|Georg Hartmann]], editor of the [[#pecham-hartmann-1542|1542 reprint]].}}

Bacon's work, although the first of the three to be written, was the last to be printed, in 1614. Pecham's ''Perspectiva Communis'', although the last to be written, spawned the largest number of manuscripts, was printed earliest (1482/3) and most often, and was clearly intended for the widest readership;&lt;ref&gt;[[#smith-2017|Smith, 2017]], p.&amp;#8239;328; Lindberg, [[#lindberg-81|1981]], pp.&amp;#8239;120–21.&lt;/ref&gt; &quot;if it were published today,&quot; says Smith, it &quot;would probably be retitled ''Perspectiva ad asinos'' or ''Optics for Dummies''.&quot;&lt;ref&gt;[[#smith-2017|Smith, 2017]], p.&amp;#8239;282.&lt;/ref&gt; Witelo's ''Perspectiva'' was printed in 1535 and reissued in 1551. In 1572 it was printed for the third time, and Alhacen's ''De&amp;nbsp;Aspectibus'' for the first time, in a single weighty volume under the title ''Opticae&amp;#8239;Thesaurus'', expertly edited—reconstructing diagrams and adding explanatory notes, citations of mathematical sources, proposition numbers and headings for Alhacen's work, and cross-references within and between the two works—by the mathematician [[w:Friedrich Risner|Friedrich Risner]].&lt;ref&gt;[[#smith-2017|Smith, 2017]], pp.&amp;#8239;328–9. Smith's translation of Alhacen ([[#smith-2001|Smith, 2001]], 2006, 2008, 2010) omits Risner's headings and uses a different section-numbering system.&lt;/ref&gt;{{efn|In the ''Opticae&amp;#8239;Thesaurus'' ([[#risner-1572|Risner, 1572]]), the two major treatises are separately paginated. Appended to Alhacen's treatise, at pp.&amp;#8239;283–8, is Ibn Mu‘ādh's&amp;#8202; essay on twilight—translated into Latin by [[w:Gerard of Cremona|Gerard of Cremona]] as ''De Crepusculis''—which was misattributed to Alhacen from the 14th century until 1967 ([[#sabra-67|Sabra, 1967]]). I have noticed that Risner's summarizing headings in Alhacen's work are also sometimes misattributed to Alhacen himself (e.g. in [[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;169, n.&amp;#8239;51, citing [[#risner-1572|Risner, 1572]], p.&amp;#8239;129, §8).}} It was Risner's edition that brought the works of Alhacen and Witelo to the attention of Benedetti and Kepler.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;498''n'',&amp;#8239;504; [[#lindberg-81|Lindberg, 1981]], p.&amp;#8239;185; [[#smith-2017|Smith, 2017]], pp.&amp;#8239;322.&lt;/ref&gt; And according to Smith, it is Risner's edition that we should blame for changing the spelling of ''Alhacen'' to ''Alhazen'' and adding Latin endings thereto.&lt;ref&gt;[[#smith-2001|Smith, 2001]], p.&amp;#8239;xxi (in the Introduction).&lt;/ref&gt;

Witelo, in the second of two postulates (&quot;''petitiones''&quot;) in Book 5 of his ''Perpectiva'', says that the location of the object-point with respect to any mirror is taken along the cathetus. He uses this postulate only to establish the cathetus rule in Prop.&amp;#8239;36: &quot;In any type of mirror, any visible point is seen on the cathetus of its incidence.&quot; For the image must be seen according to the aforesaid location of the object-point, or else it will not be seen &quot;through the mode of image&quot; (''per modum imaginis''), presumably meaning &quot;as the image of an object&quot; and not, e.g., as an independent apparition.&lt;ref&gt;[[#risner-1572|Risner, 1572]], part 2 (''Vitellonis Opticae''), pp.&amp;#8239;190,&amp;#8239;207, cited by [[#goulding-18|Goulding, 2018]], p.&amp;#8239;506; the translations and the interpretation of ''per modum imaginis'' are Goulding's.&lt;/ref&gt;

Kepler rejects Witelo's logic: &quot;First, I say that he does not do well to argue from the location of the thing seen to the location of the image, that is, out of fear that the image might cease to exist if the image should not correspond to the object in position. And indeed, in this way he would easily overturn all of catoptrics. For many things of this sort are different in the image than in the object. Next, for my part, I do not understand the postulate which he repeats from the beginning of the book,&quot; except, says Kepler, for the hint given by Alhacen ([[#alhacen-obj-img|above]]) in claiming that the state of natural things is in accordance with the situation of their principles.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;74–5.&lt;/ref&gt;

In his next proposition, Witelo repeats the cathetus rule for any type of mirror, and tries to prove it by claiming that the image of each point of an extended object must be on the cathetus in order to reproduce the size and shape of the object. But this argument is applicable only to a ''plane'' mirror, and the resulting geometric transformation of the object is not the only one that would preserve size and shape; e.g., the geometric reflection could be combined with a translation.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], p.&amp;#8239;507.&lt;/ref&gt;

On the cathetus rule for ''refraction'', Witelo faithfully recites Alhacen's argument concerning the components of motion. &quot;It is hard to see the connection,&quot; says Kepler, &quot;and even if you admit it, a mathematical deduction of what was proposed to be proved will not be forthcoming.&quot;&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;75.&lt;/ref&gt; Worse, Euclid's &quot;axiom&quot; reappears, adapted for refraction. As Kepler reports:
&lt;blockquote&gt;To Alhazen's opinion, Witelo appends the view that we had noted above as irrelevant and false in Euclid. He says, &quot;If on the surface of a transparent body a point upon which there falls a perpendicular from the seen object, happens to be hidden by the interposition of something opaque between the seen object and the point, the object will not be seen.&quot; I say that this is false. For provided that the point be free, from which the ray from the seen object to the eye is refracted, the image of the radiating object in the depth will perforce be seen.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;76. The statement in quotation marks is italicized in the original Latin edition ([[#kepler-1604|Kepler, 1604]], p.&amp;#8239;59), where it is a paraphrase rather than quote from Witelo; ''cf''. [[#risner-1572|Risner, 1572]], part 2 (''Vitellonis Opticae''), p.&amp;#8239;415.&lt;/ref&gt;
&lt;/blockquote&gt;
Thus Witelo, after striving through 400 dense pages to improve on Alhacen, regresses 15 centuries in one sentence for a last-ditch defense of the cathetus rule.

=== Benedetti: Binocularism reconsidered ===

Kepler has not been alone in his dissatisfaction with the ancient rule. In a letter to Kepler, written in late 1604 as a critique of the ''Paralipomena'', the physician Johannes Brengger proposes a modified rule, which amounts to replacing a reflective surface by its ''tangent plane'' at the point of reflection before applying the standard cathetus rule.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;533–5.&lt;/ref&gt;  A more sophisticated, independent modification is found in the optical writings of [[w:Simon Stevin|Simon Stevin]], published in 1605.&lt;ref&gt;Part of his ''Mathematical Memoirs'', first published in Dutch, then translated into Latin by Snell in 1608, and into French (more selectively) in 1608 and again in 1634 ([[#goulding-18|Goulding, 2018]], p.&amp;#8239;535; [[#dijksterhuis-55|Dijksterhuis, 1955]], pp.&amp;#8239;10,&amp;#8239;30–32, works {{serif|XIa,&amp;#8239;XIb,&amp;#8239;XIII}}).&lt;/ref&gt; Stevin's rule is a sort of binocular version of Brengger's: each eye sees a &quot;true&quot; image in the place given by Brengger; but, in a curved mirror, convergence of the lines of sight might give the ''illusion'' of a single image in a third location.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;536–43.&lt;/ref&gt; Of course, what Stevin calls an illusion is what modern readers would regard as the true location of the binocular image.

A more rigorous critic than Brengger and Stevin, and a forerunner of Kepler, is [[w:Giambattista Benedetti|Giovanni Battista Benedetti]]. His ''Book of Various Mathematical and Physical Speculations'' (Turin, 1585) contains five treatises followed by a miscellany of letters. One of the letters, addressed to a certain Conradus Terl, recognizes the role of the retina in vision, and in so doing may have anticipated [[w:Felix Platter|Felix Platter]], although Platter was first to publish.&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], pp.&amp;#8239;296–7; [[#goulding-18|Goulding, 2018]], p.&amp;#8239;512.&lt;/ref&gt; Of interest here, however, is the series of eight undated letters headed &quot;on the reflections of rays&quot; and addressed to &quot;the most excellent philosopher [[w:Francesco Vimercato|Francesco Vimercato]]&quot;,&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], pp.&amp;#8239;331–47.&lt;/ref&gt; which, according to [[w:Robert D. Goulding|Robert Goulding]], were probably written in the early 1570s.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;512–13.&lt;/ref&gt;

The first letter of the series gives several examples showing that Hero's principle of least distance does not necessarily apply to a ''concave'' mirror. In the first example,&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;513–14.&lt;/ref&gt; Benedetti shows that if we have a concave spherical mirror, with the object-point ''n'' and the observation point ''q'' on the spherical surface (extended if necessary), and seek a reflection point ''b'' opposite the chord ''qn'', the position of ''b'' is that which ''maximizes'' the path length,{{efn|Provided that the two legs of the path—from the object-point to the reflection point, and from the latter to the observation point—are constrained to be straight; if they are allowed to be curved, the path length is never a local maximum, because it can always be increased via the arc lengths of the legs (cf.&amp;nbsp;[[#born-wolf-02|Born &amp;amp; Wolf, 2002]], p.&amp;#8239;137''n''). Concerning the [[w:Fermat's principle|Hero/Fermat principle]], Goulding ([[#goulding-18|2018]], pp.&amp;#8239;513–14) makes two errors in passing. First, in his footnote 52, he fails to note that a refracted path may be a path of ''maximum'' time (again subject to the constraint that the legs are permissible ray paths) if the surface of the denser medium is sufficiently convex (consider, e.g., the refracted path through a small glass bead in the middle of the line of sight). Second, in his footnote 53, referring to the concave spherical mirror, the length of the reflected path &quot;through the unlabeled end of the diameter ''bc''&quot; is not, as he claims, the &quot;very shortest&quot; from ''q'' to ''n''; as the proposed point of reflection approaches ''q'' or ''n'', the path length approaches the length of the chord ''qn'', which is clearly shorter than the path via any other point on the sphere.}} contrary to Hero's teleological principle. Hence Benedetti prefers a ''mechanistic'' explanation of the law of reflection, which he offers in the third letter of the series.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;514–15, citing [[#benedetti-1585|Benedetti, 1585]], p.&amp;#8239;335.&lt;/ref&gt; That explanation is unconvincing by modern standards, but sets a fruitful precedent: in the same (third) letter, Benedetti goes on to seek a similarly mechanistic explanation of the cathetus rule—assuming the use of ''two'' eyes.

Benedetti is not the first optician to consider [[w:Binocular vision|binocular vision]]; Ptolemy, Alhacen, and Witelo have all confronted it.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;507–9.&lt;/ref&gt; But, whereas his predecessors have treated it as a problem—how to avoid seeing double—Benedetti treats it as an opportunity: how to perceive depth. Like Alhacen, he understands that if an object-point is to be seen singly and most distinctly, the axes of the two eyes must converge on that point; but, unlike Alhacen, he explicitly associates this convergence of the visual axes with the ''distance'' at which an object is seen singly, and he recognizes it as the mechanism of distance perception. Idiosyncratically, he adds that the distance is still perceived by looking with ''one'' eye, because (he claims) the object is still seen best when the axis of the other eye passes through it.&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], pp.&amp;#8239;335–6; [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;515–16.&lt;/ref&gt;

Armed with this new understanding of binocular vision, Benedetti considers the reflection of an object-point in a plane mirror, viewed with both eyes. Alhacen has used the cathetus rule to locate the image seen by each eye separately, and concluded that the two images coincide so that &quot;there will only be one image… and it will lie at the same place as it would if it were viewed by only one eye.&quot;&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;401–3 (Prop.&amp;#8239;4), with notes on pp.&amp;#8239;492–3, and diagrams on p.&amp;#8239;221 (other volume); ''cf''. [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;509–12 (with diagrams).&lt;/ref&gt; Benedetti inverts this reasoning: because the two lines of sight, produced through the mirror, intersect the cathetus at the same point, they intersect ''each other'' at that point, which is therefore the image—and on the cathetus. Thus, for the special case of reflection in a plane mirror, Benedetti gives the ''first valid proof of the cathetus rule'' for lines of sight other than the cathetus itself.&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], p.&amp;#8239;336; [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;516–18.&lt;/ref&gt;

For a convex spherical mirror,&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;431–2, with notes on pp.&amp;#8239;499–500, and diagrams on p.&amp;#8239;240 (other volume), pars.&amp;#8239;2.217–18.&lt;/ref&gt; and (more tersely) for a concave spherical mirror,&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;475.&lt;/ref&gt; Alhacen again relies on the cathetus rule to show that each eye sees the image-point at the same location, provided that the eyes are placed symmetrically about a plane containing the cathetus.{{efn|A statement on binocular perception of images is found at the end of Alhacen's discussion of each mirror shape, with the unexplained exception of the convex cylinder ([[#goulding-18|Goulding, 2018]], pp.&amp;#8239;511–12). For a convex conical mirror, Alhacen says that &quot;the same form and the same location for the form is perceived by each eye…; sometimes they share precisely the same location, sometimes their locations overlap, and sometimes they are separated, but only a little bit&quot; ([[#smith-2006|Smith, 2006]], p.&amp;#8239;446), where this &quot;little bit&quot; is apparently small enough to allow &quot;a single image according to sense-deduction&quot; ([[#smith-2006|Smith, 2006]], p.&amp;#8239;431). For a concave cylindrical mirror, he baldly asserts that &quot;when both eyes are looking, one image will actually form two, but they will abut or overlap, so they will appear single&quot; ([[#smith-2006|Smith, 2006]], p.&amp;#8239;481). And he gives a similar statement on what happens when a second eye is opened to each of the images formed by a concave conical mirror ([[#smith-2006|Smith, 2006]], p.&amp;#8239;485).}} In the ''concave'' case, for which Alhacen does not even offer a diagram, Benedetti gives a detailed original argument, which again avoids using the cathetus rule as a premise. Supposing at first that the object-point and both eyes are ''on'' the reflecting sphere, Benedetti shows that both (produced) lines of sight must intersect the cathetus. But only if the points of reflection are equidistant from the object-point do the intersections coincide, in which case there is a single image-point on the cathetus; otherwise, he says, the two eyes see separate images. We can see that the same reasoning applies if the eyes are moved forward, closer to the cathetus. But, as Benedetti notes, if they cross to the other side of the cathetus the object-point will be seen double and blurred (&quot;''confusè''&amp;#8202;&quot;), wherever the points of reflection may be.&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], pp.&amp;#8239;337–9; [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;518–20.&lt;/ref&gt; Moreover, he says, if the two eyes are in the same plane of reflection (confusingly called the ''surface'' of reflection), then
&lt;blockquote&gt;the place of the image will not be on the cathetus of incidence, but outside it, because the intersection of the visual axes will not be on the cathetus but outside it—and in that intersection there takes place the vision of only one image, something that the ancients did not notice.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], p.&amp;#8239;521, quoting [[#benedetti-1585|Benedetti, 1585]], p.&amp;#8239;339.&lt;/ref&gt;
&lt;/blockquote&gt;
&lt;span id=&quot;sixth&quot;&gt;Thus Benedetti ends the third letter by asserting a ''counterexample to the cathetus rule''. He does not give a proof here. In the sixth letter, however,&lt;/span&gt; he shows that a spherical burning mirror with an object-point beyond the center of curvature does not give a single focal point on the cathetus, and concludes:
&lt;blockquote&gt;Whence it follows that the convergence of reflected rays from a concave spherical mirror is not at one and the same point on the cathetus of incidence, when they are reflected from points not equidistant from the same cathetus. From this reasoning it may also be seen that what I wrote to you in the third letter is true, namely that whenever the visual axes or reflected rays are in one and the same plane of reflection, then the image of the object will in no way be seen on the cathetus of incidence in a concave spherical mirror.&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], p.&amp;#8239;343; ''cf''. [[#goulding-18|Goulding, 2018]], p.&amp;#8239;521.&lt;/ref&gt;
&lt;/blockquote&gt;
Indeed the violation of the cathetus rule in the third letter involves points of reflection that are not equidistant from the cathetus. Concerning the violation in the sixth letter, Benedetti apparently reasons that that if the reflected rays in a common plane of reflection intersect the cathetus at different points, then they must intersect ''each other'' at points ''off''&amp;#8202; the cathetus, as asserted in the third letter.

In the seventh letter (the last that deals with specular reflection), Benedetti gives another counterexample and another salvage, both for a ''convex'' spherical mirror. For the counterexample, he considers two rays from the same object-point in the same plane of reflection, and shows that if the reflected rays, when produced, intersect each other on the cathetus, then they cannot both satisfy the law of reflection.&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], pp.&amp;#8239;343–4, summarized in [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;523–5.&lt;/ref&gt; For the salvage, he takes an object-point ''b'', from which the foot of the cathetus is ''g'', and shows that if a ray from ''b'' is reflected with sufficiently glancing incidence at a point ''q'', the produced reflected ray intersects the cathetus ''bg'' in the air ''outside'' the sphere.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], p.&amp;#8239;525 &amp; Fig.&amp;#8239;15.&lt;/ref&gt; He concludes:
&lt;blockquote&gt;Therefore, if the reflected rays from the object ''b'' come to both pupils from two points of such a mirror, as distant from point ''g'' as ''q'' is, then the common point of convergence of the visual axes will be on the cathetus… where the image will appear for the reasons given above, so that this can happen not only with concave, but also with convex mirrors.&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], p.&amp;#8239;344.&lt;/ref&gt;
&lt;/blockquote&gt;
The salvage does not depend on the point of convergence being outside the sphere. It depends only on the axial symmetry about the cathetus, which implies that each produced reflected ray intersects the cathetus ''somewhere'', and that if the points of reflection are equidistant from the foot of the cathetus, so are the points of intersection.{{efn|Goulding ([[#goulding-18|2018]], p.&amp;#8239;526) explains Benedetti's conclusion thus: &quot;from his analysis of the concave mirror he extrapolated the general principle that any image location predicted by the traditional theory could be saved by the binocular theory, if the eyes were symmetrically placed on either side of the older theory's plane of reflection&quot;. I should add that the symmetry of the surface needs to be axial about the cathetus, and that the lines of sight need to be related by a rotation about the cathetus. If the symmetry were merely bilateral about &quot;the older theory's plane of reflection&quot;, it would guarantee only that the image is in that plane—not that it is necessarily on the cathetus.}}

=== Kepler: Generalized lines of sight ===

So the first disproof-and-salvage of the cathetus rule, with the first explicit counterexamples, is due to Benedetti. But here we have heard from Kepler first, because it is to him that we owe the first rebuttals of traditional ''arguments'' for the rule. Having disposed of these arguments, Kepler introduces a series of propositions of his own, &quot;''in order to make evident the true cause of the place of the image'', ignorance of which is a disgraceful stain in a most beautiful science&quot;.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;76.&lt;/ref&gt; For Kepler, as for his predecessors, an image is essentially an illusion:
&lt;blockquote&gt;''The Optical writers say it is an image, when the object itself is indeed perceived along with its colors and the parts of its figure, but in a position not its own, and occasionally endowed with quantities not its own, and with an inappropriate ratio of parts of its figure.'' Briefly, an image is the vision of some object conjoined with an error of the faculties contributing to the sense of vision. Thus, the image is practically nothing in itself, and should rather be called imagination.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;77 (Definition&amp;#8239;1); ''cf''. Malet [[#malet-1990k|1990k]], p.&amp;#8239;6.&lt;/ref&gt;
&lt;/blockquote&gt;
But what is the location of this illusory thing? In Proposition 8, Kepler eventually informs us that the distance of the image from the eye(s) is judged by triangulation, &quot;as is more amply discussed below concerning [[w:Parallax|parallaxes]]&quot;, with a baseline given by the distance between the eyes, or motion of the head, by which &quot;a single eye stands in for two that are far apart&quot;, or, at worst, the breadth of the pupil, as elaborated in Propositions 9 and 14.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;79–83.&lt;/ref&gt;{{efn|The reference to parallaxes is, I submit, an admission that a small angle of convergence between the eyes may be judged with the aid of background objects rather than by any innate ability to sense the angle.}} Thus he follows Benedetti in referring to triangulation, but goes beyond Benedetti by ''allowing baselines other than those given by binocular vision''.

Also in Proposition 9, we read that Nature intended the edges of the eyelids, and the line connecting the eyes, to be in the plane of the horizon in order to maximize the baseline for triangulation within that plane. For that reason, according to Proposition 10, when you look at an object-point via a convex mirror or &quot;the flat surface of denser media,&quot; you try to position your eyes so that the two lines of sight meet the surface at equal angles. If this condition is not met, says Kepler (again somewhat cryptically), the two lines of sight generally fail to intersect, so that you see two images, unless you strain your eyes so as to look along skew lines.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;80–81.&lt;/ref&gt; (Recall that Benedetti has noted the double vision for asymmetric placement of the eyes, but only for reflection, and only for a ''concave'' mirror.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], pp.&amp;#8239;519–20.&lt;/ref&gt;)

For cases that meet the &quot;equal angles&quot; condition, Kepler salvages the cathetus rule.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;83–6.&lt;/ref&gt; In Definition 2, he introduces the plane of reflection or refraction (again confusingly called the ''surface'' of reflection or refraction), which earlier writers have defined as the plane containing the observation point (&quot;center of vision&quot;), the object-point, and the point of reflection&amp;#10744;refraction. This plane is perpendicular to the reflecting or refracting surface (Prop.&amp;#8239;16). Now let an object-point be viewed by both eyes via a plane or spherical reflecting or refracting surface ('''Prop.&amp;#8239;17'''). For each eye, there is an point of reflection or refraction, and a line of sight (&quot;visual ray&quot;) through that point. The image-point, if one exists for the given positions of the eyes, is the point where these lines of sight meet, which must be on the line of intersection of the respective planes of reflection&amp;#10744;refraction (since these contain the lines of sight). These planes contain the object-point and are perpendicular to the surface at the respective points of reflection&amp;#10744;refraction, and hence, by the symmetry, contain the cathetus, which is therefore their line of intersection, which (as already established) contains the image-point. Thus &quot;''all the images of the seen object will be on the perpendicular from the object to the surface, whether refracting or reflecting; and this will happen to such an extent that the distance of the points of the seen object is grasped in the manner described, whether by the two eyes, or by the diameter of the breadth of one eye''.&quot;&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;86; Kepler's emphasis.&lt;/ref&gt; And it is grasped in that manner by the two eyes if the two lines of sight make equal angles with the surface (Prop.&amp;#8239;10).

Goulding initially describes Kepler's Prop.&amp;#8239;17 as a &quot;rapid proof to show that the image seen in a plane mirror would lie on the visible object's cathetus&quot;, this proof being &quot;identical to Benedetti's&quot; except in &quot;only two ways&quot;: first, Kepler does not repeat Benedetti's claim that monocular depth-perception involves the alignment of the other eye; and second, Kepler extends the argument to plane refraction. But, as Goulding adds on the same page, &quot;Kepler intended this argument to apply to any reflective or refractive surface of any shape,&quot; subject to appropriate symmetry in the placement of the eye(s).&lt;ref&gt;[[#goulding-18|Goulding, 2018]], p.&amp;#8239;529.&lt;/ref&gt; Indeed Kepler himself, in Prop.&amp;#8239;17, implicitly allows the surface to be spherical,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;86 (after the italics). But I concede that the allowance is only implicit—which may explain Goulding's incongruous conclusion that Kepler, unlike Benedetti, &quot;did not provide a proof&quot; for non-plane mirrors ([[#goulding-18|Goulding, 2018]], p.&amp;#8239;531).&lt;/ref&gt; but does not say whether it is convex or concave; his reasoning depends solely on axial symmetry about a well-defined cathetus and is otherwise indifferent to the shape of the surface or whether it is reflective or refractive.

&lt;span id=&quot;new-salvage&quot;&gt;Here I should mention a case, not mentioned by Benedetti or Kepler, in which the cathetus rule holds although the &quot;equal angles&quot; condition does not.&lt;/span&gt; Recall that [[#takahashi-defense|Takahashi defends Euclid]] by noting that if you try to look along the cathetus at the reflection of an extended object, your line of sight is blocked. Now this problem does not arise with refraction. Accordingly, consider a smooth refracting surface with the object-point on one side and your eyes on the other, with one eye (the &quot;first&quot;) on the cathetus, so that the line of sight produced from the first eye through the surface ''is'' the cathetus. If the surface and media are axially symmetrical about the cathetus, or otherwise bilaterally symmetrical about the plane of the object-point and both eyes, then, by that symmetry, the line of sight produced from the ''second'' eye through the surface intersects the cathetus. And the point of intersection is the binocular image-point.

Kepler gives his first counterexample to the cathetus rule in Proposition 18.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;86–8; ''cf''. [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;530–31.&lt;/ref&gt; Unlike Benedetti, he does not consider a concave mirror in this connection. For a ''convex'' spherical mirror, like Benedetti, he considers two rays from the same object-point in the same plane of reflection. But, whereas Benedetti supposes that the two (produced) reflected rays meet on the cathetus, and shows that they cannot both satisfy the law of reflection, Kepler supposes the law of reflection and shows by a purely geometric contradiction argument that the (produced) reflected rays meet on the observer's side of the cathetus. Indeed, as he shows more simply, the point at which they meet moves outside the sphere as we approach grazing incidence. He concludes that the cathetus rule is not universally true, &quot;unless this restriction also be added, that the sense of vision be so located with respect to the mirror as nature shows&quot;&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;88.&lt;/ref&gt;—that is, unless the lines of sight make equal angles with the surface.&lt;ref&gt;''Cf''. [[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;27,&amp;#8239;74''n''.&lt;/ref&gt; But, he adds, the departure from the cathetus is imperceptible if only one eye is used, because the lines of sight are so close together.

Kepler's theory of image location, including his disproof-and-salvage of the cathetus rule, was thought to be novel until 2018, when Benedetti's partial priority was revealed by Goulding. Kepler himself presents his theory as revolutionary, without citing Benedetti's ''Speculations''. Had he known this work, says Goulding, &quot;such an omission would have been out of character for the usually scrupulous Kepler.&quot;&lt;ref&gt;[[#goulding-18|Goulding, 2018]], p.&amp;#8239;531.&lt;/ref&gt; On that score, I can easily believe that Benedetti and Kepler independently thought of proving the cathetus rule for a plane mirror by inverting Alhacen's binocular argument, because (pardon the anecdote) [[#ingredients|so did I]], before I knew that Alhacen had introduced a second eye or a second line of sight. I can even believe that Benedetti and Kepler (unlike me) independently thought of supporting their argument by citing the same proposition {{serif|XI}}.19 of Euclid's ''Elements'', because mathematicians of bygone centuries (unlike me) knew their Euclid and cited him slavishly. Like Benedetti, Kepler gives the counterexample of the convex mirror with the two eyes in the same plane of reflection; but Goulding concedes that Kepler's treatment is &quot;more concise and elegant&quot;, and I further submit that it gives more information. Like Benedetti, Kepler rejects Hero's least-distance explanation of the law of reflection (propagated through Alhacen and Witelo), but for different reasons: the variation of the path length is negligible for reflections of stars in ponds, and the argument fails completely for refraction, supporting Kepler's claim that &quot;these operations are not those of a form that acts deliberately or keeps a goal in mind, but of matter bound to its geometrical necessities.&quot;&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;84; ''cf''. [[#goulding-18|Goulding, 2018]], pp.&amp;#8239;528–9,&amp;#8239;531.&lt;/ref&gt; There is a letter in which Kepler expresses a high opinion of Benedetti's mathematics—an opinion which, according to Goulding, he could hardly have formed from works other than the ''Speculations''.&lt;ref&gt;[[#goulding-18|Goulding, 2018]], p.&amp;#8239;531.&lt;/ref&gt; But if we accept that assessment, the evidence is still leaky because the letter dates from 16 Nov.&amp;#8239;1606, two years after the ''Paralipomena''. On that inconclusive note, I abandon this subplot and return to Kepler's treatise.

&lt;span id=&quot;first-counterex-refr&quot;&gt;In Proposition 19 of the third chapter, Kepler gives the first counterexample to the cathetus rule for ''refraction''.&lt;/span&gt; He considers a plane refracting surface, with the object-point in the denser medium and the two eyes in a common plane of refraction in the rarer medium, and shows that for sufficiently oblique incidence, the image departs from the cathetus toward the observer. He does this without knowing the exact law of refraction, by first supposing that the angle of deviation is the same for the two angles of incidence, and then showing that the departure from the cathetus is greater if, as in fact, a more oblique incidence causes a greater deviation.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;88–9; [[#goulding-18|Goulding, 2018]], p.&amp;#8239;530 &amp;amp; Fig.&amp;#8239;17. In the redrawings of Kepler's diagram by Donahue and Goulding, the incidence is not sufficiently oblique: to support the argument, point ''D''&amp;#8202; should be to the left of the (vertical) cathetus from ''E''; compare the original in [[#kepler-1604|Kepler, 1604]], p.&amp;#8239;73.&lt;/ref&gt;{{efn|In the degenerate case in which one eye is on the cathetus, the binocular image is also on the cathetus; see [[#new-salvage|above]].}}

Ending Kepler's third chapter, in Proposition 20, is the ''reductio ad absurdum'' that begins the present paper: the cathetus rule implies that we can move (e.g.) a reflected image by deforming the reflective surface in the vicinity of the cathetus while preserving it in the vicinity of the point(s) of reflection—whereas in fact, as Kepler says, &quot;it makes no difference to the place of the image, what sort of mirror surface is placed opposite the object, since the proportions of image formation are all taken from that part of the mirror upon which are the two points of reflection of light to the two eyes.&quot;&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;90.&lt;/ref&gt;{{efn|The supporting example ([[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;90–91), in which Kepler seems to have invented what we now call the [[w:Osculating circle|osculating circle]], is more sophisticated than it needs to be.}}

The imprecision of the distance of the image as judged by ''one'' eye becomes crucial in the fifth chapter of the same work, where Kepler considers a distant object seen through a glass sphere filled with water. He admits that if the eyes are sufficiently far behind the sphere, the image is seen in the air when viewed stereoscopically with two eyes,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;191, 192 (Prop.&amp;#8239;1). In modern terms, of course, this image is ''real''.&lt;/ref&gt; but is seen on the facing surface of the sphere when viewed with one eye,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;194 (Prop.&amp;#8239;6), 208–9 (Prop.&amp;#8239;17).&lt;/ref&gt; and may be seen in two places on that surface if both eyes are trained on the surface.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;195, Prop.&amp;#8239;7; ''cf''. [[#malet-1990k|Malet, 1990k]], pp.&amp;#8239;10–12 &amp;amp; Fig.&amp;#8239;5.&lt;/ref&gt; As [[w:Alan E. Shapiro|Alan E. Shapiro]] points out, this case shows that the ''perceived'' image and the ''geometrical'' image (Shapiro's terms) of the same object-point may have different locations, the former image being located by a pair of rays, and the latter by a ''pencil'' of rays (Kepler's term).&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;106–7, 124–5 &amp;amp; n.&amp;#8239;58.&lt;/ref&gt;

Later in the same chapter, Kepler considers refraction of parallel rays by a spherical surface. For deviations less than 10 degrees, using the approximation that the deviations are proportional to the angles of incidence, he shows that the refracted rays cut the axis at very nearly the same point.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;205–6 (Prop.&amp;#8239;15).&lt;/ref&gt; Then he introduces what we call the real image, which he calls a ''picture'' (Latin ''pictura''), and which, by his definition, seems to require a screen upon which it appears:
&lt;blockquote&gt;''Since hitherto an Image has been a Being of the reason, now let the figures of objects that really exist on paper or upon another surface be called pictures''.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;210 (&quot;Definition&quot;); ''cf''. [[#malet-1990k|Malet, 1990k]], p.&amp;#8239;14.&lt;/ref&gt;
&lt;/blockquote&gt;
The subsequent Propositions 20 &amp;amp; 23, which concern the picture projected by a water-filled glass sphere, imply that in order to make an intelligible picture, the rays originating from one point on the object need not converge exactly to one point in the picture; ''near''-convergence is enough. In both cases, the &quot;last intersection&quot;—&amp;#8239;that is, the limit of the intersection of the refracted ray with the axis, as the incident ray deviates less and less from the axis—is recognized as an image, implying that an image need not be perfectly stigmatic.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;211–13.&lt;/ref&gt;

But, as noted by [[w:Antoni Malet|Antoni Malet]]—against the view of previous 20th-century scholars—it is not at all clear that Kepler regards a geometrical image as acting on the eye in the same way as an object. In his ''Paralipomena''&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;192 (Prop.&amp;#8239;1).&lt;/ref&gt; and in his ''Dioptrice'' of 1611, in cases where a real image is formed in the air, Kepler conspicuously fails to invoke it in explaining what is seen by the eye(s) without a screen,&lt;ref&gt;Malet, [[#malet-1990k|1990k]], pp.&amp;#8239;5 (n.&amp;#8239;8), 21–23; [[#malet-2003|2003]], pp.&amp;#8239;118–20,&amp;#8239;134; [[#darrigol-12|Darrigol, 2012]], p.&amp;#8239;35 (Fig.&amp;#8239;1.19).&lt;/ref&gt; although he ''does'' invoke it in explaining how an upright picture can be subsequently formed on paper through another lens.&lt;ref&gt;[[#malet-2003|Malet, 2003]], p.&amp;#8239;120 &amp;amp; Figure&amp;#8239;7.&lt;/ref&gt;

These three points—
:(i) that the perceived and geometrical images may not coincide,
:(ii) that the convergence of the rays may be approximate, and
:(iii) that a geometrical image is not yet declared to be visually equivalent to an object
—are revisited later in the century.

=== Harriot and Snell: Forgotten triumph ===

Meanwhile the cathetus rule has been ironically implicated in an exasperating turn of events: the unpublished rediscoveries of the law of refraction by [[w:Thomas Harriot|Thomas Harriot]] in 1601, and [[w:Willebrord Snellius|Willebrord Snell]] in 1621.

It seems that Harriot immersed a vertical circular disk in water up to its center, sighted object-points on the rim using the center as the point of refraction, and noted that the image-points, ''when located according to the cathetus rule'', lay on a smaller ''circle'' coaxial with the disk.&lt;ref&gt;[[#lohne-59|Lohne, 1959]], pp.&amp;#8239;116–7; [[#schuster-00|Schuster, 2000]], pp.&amp;#8239;274–5.&lt;/ref&gt; It follows that the distances from the point of refraction to the object- and image-points are in a fixed ratio (the ratio of the radii of the outer and inner circles), so that the ''cosecants'' of the angles of incidence and refraction are in the same ratio, and their sines are in the inverse ratio.&lt;ref&gt;[[#goulding-22|Goulding, 2022]], p.&amp;#8239;183.&lt;/ref&gt;

Snell's surviving statement of the law begins by saying that the true ray and the apparent ray are in a fixed ratio—which is true for refraction in a plane surface, if we understand that the &quot;true ray&quot; is measured from the point of refraction to the object-point, and the &quot;apparent ray&quot; from the point of refraction to the image-point ''as located by the cathetus rule''. The statement goes on to relate the ray lengths to the cosecants of the angles.&lt;ref&gt;[[#vollgraff-1936|Vollgraff, 1936]], p.&amp;#8239;720.&lt;/ref&gt;

But the later rediscovery by [[w:René Descartes|Descartes]]—the first discovery of the law of refraction to become public—is expressed in terms of sines, not cosecants or ray lengths, and shows no other apparent influence by the cathetus rule.{{efn|The case of Descartes' co-worker [[w:Claude Mydorge|Claude Mydorge]] is less clear. Schuster ([[#schuster-00|2000]], pp.&amp;#8239;271,&amp;#8239;275–6) is impressed by the similarity between Harriot's diagram and Mydorge's, for which Goulding ([[#goulding-22|2022]], pp.&amp;#8239;191–6) offers a different explanation.}}

=== Mersenne, Roberval, Gregory: Images redefined ===

[[w:Marin Mersenne|Marin Mersenne]], in his posthumous ''L'Optique, et la Catoptrique'' (1651) edited by [[w:Gilles de Roberval|Gilles Personne de Roberval]], distinguishes between two images of the same object: the &quot;interior or sensible image&quot;, which is formed on the retina, and the &quot;exterior or apparent&quot; image, &quot;which our fantasy represents to us some place outside far or near from us, as if the object itself were in that place, from which it sends its rays to us to form the interior image…&quot;&lt;ref&gt;Quoted and translated by Shapiro, [[#shapiro-2008|2008]], p.&amp;#8239;311.&lt;/ref&gt;

Roberval, in his editorial contribution, refers to
&lt;blockquote&gt;the apparent place of the exterior image of a point of an object in all manners of vision—direct, reflected, or refracted—both for one eye alone as for two, being the point where the rays that fall on the eyes concur really or potentially (French: ''en puissance'') immediately before the eyes&amp;hellip;&lt;ref&gt;Shapiro, [[#shapiro-2008|2008]], p.&amp;#8239;295.&lt;/ref&gt;
&lt;/blockquote&gt;
In modern terms, of course, a point of &quot;potential&quot; concurrence is a ''virtual''&amp;#8202; image. Roberval also allows the rays to be ''very nearly'' concurrent,&lt;ref&gt;Shapiro, [[#shapiro-2008|2008]], p.&amp;#8239;294.&lt;/ref&gt; in agreement with Kepler on point (ii) above; but the unification of the perceived and geometrical images in the &quot;exterior&quot; image, and their visual equivalence to an object (&quot;as if the object itself were in that place…&quot;), differ with Kepler on points (i) and (iii).

[[w:James Gregory (mathematician)|James Gregory]]'s ''Optica Promota'' of 1663—chiefly known for the invention of [[w:Gregorian telescope|a reflecting telescope]] (in the Epilogue), the independent rediscovery of the law of refraction (Proposition 4),&lt;ref&gt;Discussed at length by Malet ([[#malet-1990g|1990g]]).&lt;/ref&gt; and the preface belatedly acknowledging Descartes' prior publication of this law, of which Gregory was unaware until he went to press{{efn|Gregory's ignorance of Descartes' priority is one of several pieces of evidence suggesting that the propagation of the law of refraction was slow for the first twenty years after its publication by Descartes in 1637; see [[#dijksterhuis-04|Dijksterhuis, 2004]], p.&amp;#8239;173.}}&amp;#8202;—is of interest here for its definition of an image, which is apparently independent of Mersenne and Roberval,&lt;ref&gt;Shapiro, [[#shapiro-2008|2008]], p.&amp;#8239;295.&lt;/ref&gt; and which parts with Kepler on all of points (i) to (iii) above. According to Gregory,
&lt;blockquote&gt;''An image is a similitude of a radiating body, arising from the divergence or convergence of the rays belonging to individual points of the radiating body, from individual points or to individual points of a single surface.&lt;ref&gt;&quot;''Imago est similitudo materiæ radiantis, orta ex divergentiâ, vel convergentiâ radiorum, singulorum materiæ radiantis punctorum, a punctis singulis, vel ad puncta singula unius superficiei.''&quot; —&amp;#8239;[[#gregory-1663|Gregory, 1663]], p.&amp;#8239;1 (Definition 9).&lt;/ref&gt;''
&lt;/blockquote&gt;
This definition, like Roberval's, allows no distinction between perceived and geometrical image-points and applies to both binocular and monocular viewing,&lt;ref&gt;[[#gregory-bruce-06|Gregory/Bruce, 2006]], Props.&amp;#8239;28,&amp;#8239;29,&amp;#8239;36; [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;128–30.&lt;/ref&gt; and attributes the &quot;similitude&quot; to the defining feature of a geometrical image: the convergence or divergence of the rays. But, unlike Roberval (and Kepler), Gregory does not allow the point of divergence or convergence to be an approximation or limiting case; concerning the image of an object-point ''B'' seen by reflection, Gregory writes:
&lt;blockquote&gt;From the points of the pupil [''A''], draw through the points of reflection all the lines of reflection, in whose concourse ''L'' (provided they concur) will be the apparent place of the image of the point ''B''. If, however, they do not concur in one point, no distinct and fixed place of the image of the visible point ''B'' will exist.&lt;ref&gt;End of Prop.&amp;#8239;36, as translated by Shapiro ([[#shapiro-1990|1990]], p.&amp;#8239;129).&lt;/ref&gt;
&lt;/blockquote&gt;
Although the diagram supporting this statement shows the point of concourse as being behind the mirror (giving a virtual image), the wording is equally applicable if the point is in front (giving a real image). Moreover, the initial statement of the problem indicates that the solution should be equally applicable to refraction,&lt;ref&gt;[[#gregory-bruce-06|Gregory/Bruce, 2006]], Prop.&amp;#8239;36.&lt;/ref&gt; which it is. And indeed the initial definition is applicable to both real and virtual images, and to both reflection and refraction. Gregory's insistence on exact concurrence may look like a loss of generality, but is understandable in view of his coverage of the exact imaging properties of conic sections in reflection and refraction.&lt;ref&gt;''Cf''. Malet, [[#malet-1990g|1990g]].&lt;/ref&gt;

Mersenne, Roberval, and Gregory have not addressed the cathetus rule directly; but their refinement of the concept of the image will be pivotal in a high-profile case.

=== Tacquet: Affirmation and exception ===

According to Malet:
&lt;blockquote&gt;By the late sixteenth century it was a well-known fact that [distant] things perceived through convex lenses appear inverted or upright according to the distance from the eye to the lens. Empirical accounts of the properties of convex lenses, such as [[c:William Bourne (mathematician)|William Bourne]]'s 'Treatise on the properties and qualities of glasses for optical purposes' (1585),&lt;ref&gt;Printed in [[#halliwell-1839|Halliwell, 1839]], pp.&amp;#8239;32–47. &quot;1585&quot; is [[w:Albert Van Helden|Van Helden]]'s dating of the treatise, whereas [[w:Sven Dupré|Dupré]] dates it to 1579/80 ([[#dupre-10|Dupré, 2010]], pp.&amp;#8239;137–8).&lt;/ref&gt; did not fail to mention that&amp;#8239; (1) when the eye is removed from the lens beyond the 'burnynge beame', or focus, all [distant] things seen through the lens appear inverted, and&amp;#8239; (2) when the eye lies between the burning focus and the lens all things seen through the lens appear upright and enlarged, and the more so the closer the eye to the focus.&lt;ref&gt;[[#malet-2003|Malet, 2003]], p.&amp;#8239;116; &quot;[distant]&quot; is my addition, for context. Compare Bourne, chapters {{serif|VI}} to {{serif|VIII}}, in [[#halliwell-1839|Halliwell, 1839]], pp.&amp;#8239;42–4. On the contrivance mentioned at the end of chap.&amp;#8239;{{serif|VI}} and elaborated in chap.&amp;#8239;{{serif|IX}}, see [[#dupre-10|Dupré, 2010]].&lt;/ref&gt;
&lt;/blockquote&gt;
Here we are chiefly interested in Malet's point (2), under which we should also note that when the eye reaches the focus, as Bourne says, &quot;yow shall discerne nothinge thorowe the glasse: But like a myst, or water&quot;.&lt;ref&gt;[[#halliwell-1839|Halliwell, 1839]], p.&amp;#8239;44.&lt;/ref&gt;

Kepler explains point (2) in his ''Dioptrice''. He shows that when an object-point is viewed through a convex lens at such a distance that the refracted rays converge toward another point, with the eye between that point and the lens, the object is seen upright (Proposition 70) and blurred (&quot;''confusa''&quot;), the more blurred as the eye is further from the lens, since the convergence is greater (Prop.&amp;#8239;71), and most blurred when the eye reaches the point of convergence (Prop.&amp;#8239;74). Moreover the image is magnified (Prop.&amp;#8239;80), and the more so as the eye recedes from the lens toward the point of convergence (Prop.&amp;#8239;82).&lt;ref&gt;[[#kepler-1859|Kepler (1859)]], pp.&amp;#8239;542–7. The location of these passages was assisted by Darrigol ([[#darrigol-12|2012]], pp.&amp;#8239;34–5), Malet ([[#malet-2010|2010]], pp.&amp;#8239;283–6), Shapiro ([[#shapiro-1990|1990]], p.&amp;#8239;160 &amp;amp; n.&amp;#8239;184), and ''translate.google.com''. On Kepler's explanation of Prop.&amp;#8239;82, see [[#malet-2003|Malet, 2003]], p.&amp;#8239;114 &amp;amp; Figure 4. Props.&amp;#8239;80&amp;#8239;&amp;amp;&amp;#8239;82 are used in Kepler's subsequent explanation of the magnifying power of a Dutch telescope; see Malet, [[#malet-2003|2003]] at p.&amp;#8239;122, or [[#malet-2010|2010]] at p.&amp;#8239;286.&lt;/ref&gt;

Gregory, in the following passage, confirms the blur but is indifferent to whether the convergence is caused by a lens or a mirror:
&lt;blockquote&gt;''Corollary 4.''

…&amp;#8239;[I]f the rays from one point converge toward another point behind the eye [''post oculum''], no place can be assigned to this point except (if we will) behind the eye at the concourse of the rays: hence the image formed of such points may conveniently be called an image behind the eye.

&lt;span id=&quot;gregory-prop-30&quot;&gt;'''Prop.&amp;#8239;30. Theorem.'''&lt;/span&gt;

''With the rays from one point converging toward a point situated behind the eye, it is impossible to make distinct vision.''

For every eye is so constructed as to see distinctly either remote [points], which radiate as if in parallel, or near ones, which send out diverging rays; but in no eye is the retina distinctly painted by the converging rays (which originate from artifice and not from nature), because the crystalline humor{{efn|That is, the lens.}} gathers [''congregat''] these rays into a point in the vitreous humor, and sends them disgregated to the retina, from which disgregation arises blurred vision—as shown by Kepler.&lt;ref&gt;Translated from [[#gregory-1663|Gregory, 1663]], p.&amp;#8239;41, and in some places differing from [[#gregory-bruce-06|Gregory/Bruce, 2006]].&lt;/ref&gt;
&lt;/blockquote&gt;
This &quot;image behind the eye&quot; is what we would now call a '''virtual object''' presented to the eye. Although there is no mention of the cathetus rule here, the ''mirror version of the same experiment''—in which rays converge from a concave mirror toward a point behind the eye—is the only case in which the cathetus rule is ''not'' upheld by [[w:André Tacquet|André Tacquet S.J.]] in his ''Catoptrica Tribus Libris Exposita'' (Catoptrics explained in three books), posthumously published in 1669. At the end of Book 1, Tacquet says of the cathetus rule:
&lt;blockquote&gt;This theorem is the most fruitful of all of catoptrics, whereby nearly all the phenomena of plane and convex mirrors are demonstrated, as will become evident from all of book two and book three. Consequently, its truth is in turn extraordinarily established: for it cannot be false, since it agrees wonderfully with all phenomena without exception.&lt;ref&gt;[[#tacquet-1669|Tacquet, 1669]], p.&amp;#8239;223, quoted in translation by Shapiro ([[#shapiro-1990|1990]], p.&amp;#8239;144).&lt;/ref&gt;
&lt;/blockquote&gt;
But he immediately adds:
&lt;blockquote&gt;''Whether and when this proposition has a place with concave mirrors will be plain from what is to be said in Book 3''.&lt;ref&gt;[[#tacquet-1669|Tacquet, 1669]], p.&amp;#8239;223, italics in the Latin.&lt;/ref&gt;
&lt;/blockquote&gt;
And in Book 3, just before Proposition 22,&lt;ref&gt;[[#tacquet-1669|Tacquet, 1669]], p.&amp;#8239;256.&lt;/ref&gt; he warns that &quot;in concave ones we postulate this only for the moment, until the extent of its truth becomes apparent.&quot; In Props.&amp;#8239;29&amp;#8239;&amp;amp;&amp;#8239;30,&lt;ref&gt;[[#tacquet-1669|Tacquet, 1669]], p.&amp;#8239;259.&lt;/ref&gt; he comes to the experiment just mentioned, in which the eye intercepts converging rays from a concave mirror. Here the cathetus rule locates the image ''behind'' the eye—in agreement with Gregory's terminology—whereas the mind inevitably construes any visible image as being ''in front'' of the eye, leading Tacquet to conclude:
&lt;blockquote&gt;''Therefore Alhazen, Witello, and other opticians following them err in considering that just as in plane and convex mirrors so in concave ones the image never appears outside the intersection of the reflected ray with the cathetus of incidence.''
&lt;/blockquote&gt;
The quote is translated by Shapiro,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;172, n.&amp;#8239;107; italics in the Latin.&lt;/ref&gt; who further reports that as late as 1735, [[w:Samuel Clarke|Samuel Clarke]] faulted Tacquet for making even that exception to the cathetus rule,&lt;ref&gt;[[#rohault-clarke-1735|Rohault/Clarke, 1735]], p.&amp;#8239;278''n''.&lt;/ref&gt; while [[w:Christian Wolff (philosopher)|Christian Wolff]] upheld the rule for two eyes provided that they were not in the same plane of incidence.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;172, n.&amp;#8239;108.&lt;/ref&gt; In allowing the eyes to be asymmetrically placed in different planes of incidence, Wolff's proviso is too permissive—as Benedetti and Kepler knew.

=== Barrow &quot;destroys&quot; the doctrine ===

The Rev. [[w:Isaac Barrow|Isaac Barrow]], inaugural [[w:Lucasian Professor of Mathematics|Lucasian Professor]] at Cambridge, in the first of his ''Lectiones {{serif|XVIII}}'' (Eighteen Lectures) published in 1669, defines images thus:
&lt;blockquote&gt;…&amp;#8239;Images are clearly nothing other than light from objects so reflected or refracted that it is again collected in one place and in such a situation as it had when it flowed from the original object and proceeded in a direct path to the eye; whereby it happens that images represent objects similarly but as if they were located elsewhere.&lt;ref&gt;''Lectiones'' {{serif|I}}:5 ([[#barrow-1669|Barrow, 1669]], p.&amp;#8239;4, quoted in translation by Shapiro, [[#shapiro-1990|1990]], p.&amp;#8239;107).&lt;/ref&gt;
&lt;/blockquote&gt;
In the third lecture he reprises the idea:
&lt;blockquote&gt;Indeed by the term ''image'', I understand nothing but the place from which a number of rays (as many as suffice to affect vision) seem to diverge or spread in the same manner as when they are diffused by primary objects.&lt;ref&gt;''Lectiones'' {{serif|III}}:16 ([[#barrow-1669|Barrow, 1669]], p.&amp;#8239;30), cited (not translated) by Shapiro ([[#shapiro-1990|1990]], p.&amp;#8239;166, n.&amp;#8239;6); my italics.&lt;/ref&gt;
&lt;/blockquote&gt;
As Shapiro explains,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;106–7 (&amp;amp; n.&amp;#8239;5), 124–5,&amp;#8239;165.&lt;/ref&gt; Barrow's principle of image location, which was rightly linked to him in the 18th century, was wrongly credited to Kepler in the 20th. In fact Barrow agrees with Roberval: he follows Roberval and Gregory, against Kepler, by strictly equating the perceived and geometrical images, and by recognizing the manner in which an image imitates an object; but, as we shall see, he follows Kepler and Roberval, against Gregory, by not requiring an image to be strictly stigmatic.

The case of the eye intercepting converging rays, whether from a convex lens as in Kepler's example, or from a concave mirror as in Tacquet's, is known as the '''Barrovian case'''&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;144,&amp;#8239;159–65.&lt;/ref&gt; because it is taken up by Barrow—citing Tacquet but, strangely, not Kepler in this connection—at the end of his lectures; the relevant passage has been translated from the Latin by [[w:George Berkeley|Berkeley]] and, independently, by Clarke.&lt;ref&gt;[[#berkeley-1901|Berkeley (1901)]], pp.&amp;#8239;137–40; [[#rohault-clarke-1735|Rohault/Clarke, 1735]], pp.&amp;#8239;260–61''n''.  Fay's recent translation of all eighteen lectures ([[#barrow-fay-87|Barrow/Fay, 1987]]) is apparently out of print.&lt;/ref&gt; Here Barrow notes that because diverging rays appear to come from a finite distance, and parallel rays from an infinite distance, converging rays ought to appear to come from beyond infinity,&lt;ref&gt;[[#berkeley-1901|Berkeley (1901)]], p.&amp;#8239;138.&lt;/ref&gt; whereas in fact, in the case in question, the image may seem closer than the object, and certainly seems to come closer as the rays become more convergent&lt;ref&gt;Shapiro ([[#shapiro-1990|1990]], p.&amp;#8239;160, line 6) erroneously has &quot;divergence&quot; instead of &quot;convergence&quot;.&lt;/ref&gt;—that is, as the eye recedes toward the point of convergence—until &quot;the object appearing extremely near begins to vanish into mere confusion.&quot;&lt;ref&gt;[[#berkeley-1901|Berkeley (1901)]], p.&amp;#8239;139.&lt;/ref&gt; Indeed the image seems to come closer because (as mentioned by Kepler but not Barrow) the magnification increases, and because (as mentioned by neither, but easily observed) the direction of the image becomes more sensitive to sideways movement of the eye—although the apparent movement of the image is the wrong way for an image in ''front'' of the eye. As Barrow notes, the looming of the image offends not only &quot;our Notion&quot; (his principle of image location), but also &quot;that antient and common one&quot; (the cathetus rule):
&lt;blockquote&gt;It seems so much to overthrow that antient and common one, which is more a-kin to ours than any other, that the learned Tacquett was forced by it to renounce that Principle, (upon which alone, almost all his Catoptricks depend) as uncertain, and not to be depended upon, whereby be overthrew his own Doctrine.&lt;ref&gt;[[#rohault-clarke-1735|Rohault/Clarke, 1735]], p.&amp;#8239;261''n''.&lt;/ref&gt;
&lt;/blockquote&gt;
After this caricature of Tacquet's position, Barrow immediately concedes:
&lt;blockquote&gt;Which, nevertheless, I do not believe he would have done, had he but considered the whole matter more thoroughly, and examined the difficulty to the bottom.&lt;ref&gt;[[#berkeley-1901|Berkeley (1901)]], p.&amp;#8239;139; this statement is elided in Clarke's translation.&lt;/ref&gt;
&lt;/blockquote&gt;
The concession is startling—the more so for its want of explanation—in that it seems to imply that Tacquet's purported counterexample to the cathetus rule is ''not'' a counterexample. That indeed is the position subsequently taken by Clarke, who argues that the cathetus rule is not in play, because the reflected rays, being intercepted by the eye, do not meet the cathetus.&lt;ref&gt;[[#rohault-clarke-1735|Rohault/Clarke, 1735]], p.&amp;#8239;278''n''.&lt;/ref&gt; In his commentary on the Barrovian passage, Clarke explains the apparent closeness of the image by noting that&amp;#8239; (i) if the eye is sufficiently close to the point of convergence, we cannot simultaneously train both eyes on the object-point through the glass (however large it may be), and with only one eye the judgment of distance is inferior and influenced by the proximity of the glass, and&amp;#8239; (ii) as the eye recedes, the increasing magnification (and brightness, in the case of a luminous object) makes the image seem to come closer.&lt;ref&gt;[[#rohault-clarke-1735|Rohault/Clarke, 1735]], p.&amp;#8239;262''n''.&lt;/ref&gt; Berkeley's explanation,&lt;ref&gt;[[#berkeley-1901|Berkeley (1901)]], pp.&amp;#8239;140–43 (§§&amp;#8239;31,&amp;#8239;35–6); ''cf''. [[#cardona-gutierrez-20|Cardona &amp;amp; Gutierrez, 2020]].&lt;/ref&gt; although earlier, is more modern, noting that that the convergence of rays via a lens or mirror is not the only reason why an object may appear blurred; another is that the object is too ''close''!

A late twist in the story of the Barrovian case—presumably unknown to all the characters from Bourne in the 16th century to Clarke in the 18th—is that the concave-mirror version, including the application of the cathetus rule, is discussed in Ptolemy's ''Optics''.&lt;ref&gt;Experiment {{serif|IV}}.1, translated in [[#smith-1996|Smith, 1996]], pp.&amp;#8239;194–5, with further commentary in [[#smith-2017|Smith, 2017]], pp.&amp;#8239;104–7.&lt;/ref&gt; For a given position of the eye and a given point of reflection, Ptolemy marks three object positions for which the cathetus rule will place the image respectively at the eye, behind the eye, and nowhere (or, as we would say, at infinity), and indicates the range of object positions for which the rule places the image behind the mirror. For the case in which the rule would place the image behind the eye, he claims that the object seems to be in front of the mirror (in violation of the rule) because the visual faculty is biased toward the surface from which the reflection comes. Similarly, when the rule places the image at infinity or at the eye, Ptolemy says it is perceived to be ''on the mirror''. Later, for a single spherical surface, Ptolemy gives what would amount to a refractive version of the experiment, if it were described in the same detail.&lt;ref&gt;Theorem {{serif|V}}.9, translated in [[#smith-1996|Smith, 1996]], p.&amp;#8239;252, with commentary in [[#smith-2017|Smith, 2017]], p.&amp;#8239;119 &amp;amp; figure&amp;#8239;3.15.&lt;/ref&gt; Less likely to have escaped notice is the related example given by Alhacen,&lt;ref&gt;[[#smith-2006|Smith, 2006]], p.&amp;#8239;451 (par.&amp;#8239;2.331) and figure 5.2.34b on p.&amp;#8239;254 (other volume); [[#risner-1572|Risner, 1572]], p.&amp;#8239;162, reprised by Witelo at his pp.&amp;#8239;314–5.&lt;/ref&gt; and cited by Bacon,&lt;ref&gt;[[#bacon-combach-1614|Bacon/Combach, 1614]], pp.&amp;#8239;139–40; [[#bacon-burke-1928|Bacon/Burke, 1928]], pp.&amp;#8239;553–5; [[#smith-2017|Smith, 2017]], p.&amp;#8239;268.&lt;/ref&gt; for which the cathetus rule places one of the images behind the eye. Here Alhacen does not comment on the evident impossibility, whereas Bacon, like Ptolemy, blames the limitations of vision:
&lt;blockquote&gt;But in all these diversities of appearances the image is never truly apprehended unless its place is beyond the mirror, or between the sight and the mirror; hence what appears in the center of the eye or behind the head is not perceived there. For vision is not born to apprehend the positions of forms unless they are in front of it.&lt;ref&gt;[[#bacon-combach-1614|Bacon/Combach, 1614]], p.&amp;#8239;140; ''cf''. [[#bacon-burke-1928|Bacon/Burke, 1928]], p.&amp;#8239;555.&lt;/ref&gt;
&lt;/blockquote&gt;

In the Barrovian case, in the words of Barrow's definitions of an image, the point toward which the rays converge is neither &quot;light… again collected in one place&quot;, because the light never gets there, nor a place from which rays &quot;seem to diverge&quot;, because they ''con''verge. (That is, in modern terms, it is neither a real image nor a virtual image.) Therefore, according to Barrow's criteria, it should not be the perceived image. But what should be? Barrow does not have an answer that passes the test of experiment. So we are forced to admit that in the Barrovian case, as in all the other cases surveyed by Tacquet (if he is to be believed), the ancient cathetus rule does no worse than Barrow's post-Keplerian principle of image location.{{efn|In modern terms, the point toward which the rays converge in the Barrovian case is a virtual object presented to the front surface of the eye, which refracts the rays toward a nearer point, which in turn becomes a virtual object presented to the interface between the cornea and the aqueous humor, and so on, until a real image is formed in front of the retina. From this image the rays diverge again to form a blurred picture ''on'' the retina (as Gregory notes in his [[#gregory-prop-30|Prop.&amp;#8239;30]], quoted above). What is presented to the observer's retina is thus easily explained and uncontroversial. What the observer makes of it is another matter: &quot;Insofar as I can determine&quot;, says Shapiro ([[#shapiro-1990|1990]], p.&amp;#8239;178, n.&amp;#8239;206), &quot;there is still no generally accepted explanation for the 'Barrovian case.'&amp;#8239;&quot;}}

However, Barrow's principle manifestly does better than &quot;that antient and common one&quot; in explaining another case: the location of the image seen by refraction in a plane surface, which Barrow determines by some inspired pre-calculus geometry and &quot;the most recently given law or hypothesis of refraction (discovered by the illustrious Descartes, but now, I believe, embraced by most of the better Opticians…)&quot;.&lt;ref&gt;Translated from [[#barrow-1669|Barrow, 1669]] (introduction); ''cf''. [[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;113.&lt;/ref&gt;

[[File:Barrow-tangential.svg|thumb|374px|Isaac Barrow's location of the tangential image ''Z'' of an object-point ''A'' seen by an observer at ''O'' due to refraction. The tangential image is the point of tangency between the refracted ray produced back from ''O'', and the ''caustic'' (common tangent curve) of all the other produced refracted rays from the same object-point in the same plane of refraction. Point ''K'' is the image location given by the old cathetus rule; it lies on the cathetus ''AB''. Point ''P'', where the caustic meets the cathetus, is the ''paraxial'' image, i.e. the image of ''A'' seen by an observer on the cathetus, below ''B''. (Diagram by the author, after Barrow.)]]

Given an object-point ''A'' in the rarer medium, another point ''X'' in that medium, and the constraint that the (produced) refracted ray must pass through ''X'', Barrow seeks the refracted ray. He finds that there are two solutions which merge under a certain condition, under which he renames ''X'' as ''Z'' and supposes that the eye (at ''O'') looks along the refracted ray, which thereby becomes what he calls the &quot;principal ray&quot; (''ZO'' in the figure), i.e. the ray through the center of the eye.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;130,&amp;#8239;132–3.&lt;/ref&gt; He then argues that ''Z'' is where the eye sees the image, because if we take two neighboring refracted rays from the same object-point ''A'' in the same plane of refraction, one on each side of the principal ray (e.g., the rays passing through ''C'' and ''D''), and produce them back through the interface, they intersect the principal ray ''ZO'' on opposite sides of the point ''Z''. And this point, as he has found, is ''beyond the cathetus'' with respect to the eye.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;133–4.&lt;/ref&gt; Whereas &quot;Alhazen and most of the multitude of opticians after him&quot; would place the image at ''K'', i.e. at the intersection of the produced principal ray ''ZO'' and the cathetus ''AB'', Barrow notes that only one ray from ''A'' (namely ''AO'') is produced back through ''K'' unless the eye is on the cathetus,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;134 &amp;amp; n.&amp;#8239;86, quoting ''Lectiones'' {{serif|V}}, §21, incorrectly numbered 20 in the original printing ([[#barrow-1669|Barrow, 1669]], pp.&amp;#8239;44–6).&lt;/ref&gt; in which case, as he shows in the previous lecture,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;131–2.&lt;/ref&gt; all refracted rays that enter the pupil will, when produced back through the interface, intersect the cathetus at nearly the same point (marked ''P'' in our figure).

If the eye is ''off''&amp;#8202; the cathetus, the image-point ''Z''&amp;#8202; found by Barrow&amp;#8202; is what we now call the '''tangential''' image—because it is the point of tangency between the (produced) line of sight and the '''[[w:Caustic (optics)|caustic]]''' (common tangent curve) of all the (produced) refracted rays originating from the same object-point in the same plane of refraction.&lt;ref&gt;[[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;73–4; [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;109,&amp;#8239;139. The term ''caustic''—but not the concept—was apparently coined in 1690 by [[w:Ehrenfried Walther von Tschirnhaus|Ehrenfried Walther von Tschirnhaus]] ([[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;28,&amp;#8239;74–5; [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;157–8 &amp;amp; n.&amp;#8239;165).&lt;/ref&gt; This tangency explains his procedure: for the given object-point ''A'', there can be two refracted rays produced through the target point ''X''&amp;#8202; if ''X'' is ''off'' the caustic, but only one if it is ''on'' the caustic.&lt;ref&gt;''Cf''. [[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;108, Figure 1.&lt;/ref&gt;

If, on the contrary, the eye is ''on'' the cathetus (below ''B''), the image-point found by Barrow is the cusp of the caustic (our point ''P''), which is now known as the '''[[w:Paraxial approximation|paraxial]]''' image, and which ''satisfies the cathetus rule in the limiting case''.{{efn|Barrow finds the paraxial image before he finds the tangential image. That the former is the limit of the latter follows from the displayed equation on p.&amp;#8239;148 of [[#shapiro-1990|Shapiro, 1990]], by letting {{mvar|i }}and{{mvar| r}} approach zero, so that their cosines approach 1, yielding the paraxial equation on p.&amp;#8239;147. These equations are for a spherical surface, but are easily adapted for a plane surface by putting ''&amp;rho;''&amp;#8239;&amp;rarr;&amp;#8239;&amp;infin;.}} Barrow refers to the tangential image as the &quot;relative&quot; image, which is &quot;mutable&quot; and &quot;less important&quot;, and to the paraxial image as the &quot;absolute&quot; image, which is &quot;simple&quot; and &quot;principal&quot;.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;109,&amp;#8239;136.&lt;/ref&gt; In both cases he applies the term &quot;image&quot; to a point that ''nearly'' coincides with all the intersections between rays entering the pupil from the same object-point; in this he follows Kepler and Roberval, against Gregory.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;128–30; [[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;211–13 (Props.&amp;#8239;20,&amp;#8239;23); [[#gregory-bruce-06|Gregory/Bruce, 2006]], Prop.&amp;#8239;36.&lt;/ref&gt;

Nowadays we tend to think of the tangential image in contradistinction to the '''sagittal''' image. The latter, Barrow ignores;&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;172, n.&amp;#8239;101.&lt;/ref&gt; where he says that only one (produced) refracted ray passes through the image-point alleged by the cathetus rule (point ''K''), he implicitly confines his attention to rays in the same plane of refraction on the same side of the cathetus. It is left to his successor and former student, [[w:Isaac Newton|Isaac Newton]], to point out that in consequence of the axial symmetry about the cathetus, a whole cone of refracted rays shares this property, giving a second image-point (''K''), which is now called the sagittal image, and which ''exactly satisfies the cathetus rule''.&lt;ref&gt;[[#newton-anon-1728|Newton/anon., 1728]], pp.&amp;#8239;104–5 (scholium) &amp;amp; Plate 7; [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;135–6,&amp;#8239;172.&lt;/ref&gt; Recall, however, that Newton's observation is partly anticipated by Kepler, who considers two rays in the said cone,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;85–6 (Prop.&amp;#8239;17); ''cf''. [[#darrigol-12|Darrigol, 2012]], p.&amp;#8239;74, and [[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;121.&lt;/ref&gt; but subsequently ignores the sagittal image.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;123–4.&lt;/ref&gt;

Interchanging the dense and rare media, we return to the [[#first-counterex-refr|case considered by Kepler]] in which (e.g.) one looks into still water from above, with the eyes in a common plane of refraction.&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;88–9 (Prop.&amp;#8239;19).&lt;/ref&gt; Here Barrow offers the following &quot;not inelegant&quot; experiment, which confirms the proposition of Kepler (not cited) and &quot;clearly destroys the doctrine of Alhazen and his followers&quot;.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;134–5 &amp;amp; n.&amp;#8239;89, quoting ''Lectiones'' {{serif|V}}, §22, incorrectly numbered 21 in the original printing ([[#barrow-1669|Barrow, 1669]], p.&amp;#8239;46).&lt;/ref&gt; Attach a weight ''F''&amp;#8202; to a string and hang it from a pivot ''G'', with ''G'' above the water's surface and ''F''&amp;#8202; below, adjusting the height and depth so that, when your eyes are level and facing the string, the refracted image of ''F''&amp;#8202; appears just below the reflected image of ''G''. With your eyes in this natural attitude, the two images indeed appear aligned with the string and its reflected image—that is, on the cathetus. But now tilt your head so that both eyes are in a common plane of reflection&amp;#10744;refraction, and the refracted image of ''F''&amp;#8202; has moved toward you, away from the reflected image of ''G''—that is, away from the cathetus, in defiance of the ancient rule. Seeing is believing.{{efn|Yes, I ''did'' try this at home.}}

For oblique reflection in a convex spherical mirror, Barrow's &quot;relative&quot; image, like Kepler's image with the eyes in a common plane of reflection,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;86–8 (Prop.&amp;#8239;18).&lt;/ref&gt; is on the observer's side of the cathetus. Considering the object-point as a general point on an infinitely long line perpendicular to the mirror, Barrow shows that the image of the line is curved and angled to it, whereas the cathetus rule, &quot;gratuitously assumed and contrary to reason&quot;, would have the image in line with the object. But, in an apparent reference to Tacquet—who claims to have verified experimentally &quot;a hundred times&quot; that the image is in line, and backs the claim by appealing to the axial symmetry about the cathetus,&lt;ref&gt;[[#tacquet-1669|Tacquet, 1669]], p.&amp;#8239;222 (Prop.&amp;#8239;19).&lt;/ref&gt; although the line of sight violates that symmetry—Barrow concedes that the deviation of this image from the cathetus is harder to observe than the deviation of the refracted image in the aforesaid plumb-line experiment, with the eyes in a common plane of refraction: there the reflected image marks the cathetus, and the refracted image is manifestly not on it.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;142–3, quoting Barrow, ''Lectiones'' {{serif|XVI}}.&lt;/ref&gt;

=== Newton and the &quot;axiom&quot; of stigmatism ===

Newton's salvage of the cathetus rule for the sagittal image, in the case of axial symmetry about the cathetus, is relegated to his posthumously published ''Optical Lectures''.&lt;ref&gt;[[#newton-anon-1728|Newton/anon., 1728]], pp.&amp;#8239;104–5 (scholium) &amp;amp; Plate 7; [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;135–6.&lt;/ref&gt; In his better-known ''Opticks'', the first 19 pages consist of eight definitions followed by eight &quot;Axioms and their Explications&quot;, by which he then claims to have given &quot;the sum of what hath hitherto been treated of in Opticks&quot; or at least &quot;what hath been generally agreed on&quot;.&lt;ref&gt;[[#newton-2010|Newton (2010)]], pp.&amp;#8239;19–20.&lt;/ref&gt;

&quot;Despite his grandiose claim,&quot; says Shapiro,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;149.&lt;/ref&gt; &quot;he did do a remarkable job of compressing elementary geometrical optics into nine pages.&quot; The compression begins with the following &quot;axiom&quot; on p.&amp;#8239;10:
&lt;blockquote&gt;'''Ax. {{serif|VI}}.'''

''Homogeneal Rays which flow from several Points of any Object, and fall perpendicularly or almost perpendicularly on any reflecting or refracting Plane or spherical Surface, shall afterwards diverge from so many other Points, or be parallel to so many other Lines, or converge to so many other Points, either accurately or without any sensible Error. And the same thing will happen, if the Rays be reflected or refracted successively by two or three or more Plane or Spherical Surfaces''.

The Point from which Rays diverge or to which they converge may be called their ''Focus''.&amp;#8239;…
&lt;/blockquote&gt;
In other words, for reflection or refraction by a plane or spherical surface, if the angles of incidence are not too large, the image of the object-point (although the term ''image'' has not yet been introduced) will be near enough to ''stigmatic'', at least for &quot;homogeneal&quot; (monochromatic) rays. This axiom leads to four rules, stated without proof, for locating the focus of the rays reflected or refracted by a plane surface (&quot;''Cas''.&amp;#8239;1&quot;), reflected by a spherical surface (&quot;''Cas''.&amp;#8239;2&quot;), refracted by a spherical surface (&quot;''Cas''.&amp;#8239;3&quot;), and refracted by a lens (&quot;''Cas''.&amp;#8239;4&quot;). Here we should emphasize, although Newton does not, that in the first three cases—those which involve a single surface and a single cathetus—the stated location of the focus is ''on the cathetus''.

In his next &quot;axiom&quot; (p.&amp;#8239;14), Newton gives the condition under which a set of foci makes a picture; but, unlike Kepler, he implicitly acknowledges the independent existence of the foci:
&lt;blockquote&gt;'''Ax. {{serif|VII}}.'''

''Wherever the Rays which come from all the Points of any Object meet again in so many Points after they have been made to converge by Reflection or Refraction, there they will make a Picture of the Object upon any white Body on which they fall''.
&lt;/blockquote&gt;
Thence he explains the [[w:Camera obscura|camera obscura]], the eye, long- and short-sightedness, and correcting spectacles.

In the final &quot;axiom&quot; of the set (p.&amp;#8239;18), he endorses Barrow's principle of image location without naming Barrow or using the word ''image'':
&lt;blockquote&gt;'''Ax. {{serif|VIII}}.'''

''An Object seen by Reflexion or Refraction, appears in that place from whence the Rays after their last Reflexion or Refraction diverge in falling on the Spectator's Eye''.
&lt;/blockquote&gt;
For a plane mirror, he explains, if that place of divergence is point ''a'', &quot;these Rays do make the same Picture in the bottom of the Eyes as if they had come from the Object really placed at ''a''…&quot; As further examples he cites a prism with refracted rays diverging from ''d'', and a lens with refracted rays diverging from ''q''. Then he abruptly refers to the &quot;Image of the Object&quot; at ''q''&amp;#8202; as having a certain size, and goes on to use the term ''image'' routinely, without further introduction. But he has implied, immediately after Ax.&amp;#8239;{{serif|VI}}, that a place of divergence is a &quot;focus&quot;, allowing us to interpret that &quot;axiom&quot; as giving sufficient conditions for the approximate stigmatism of the image.

Now let us consider the implications of stigmatism. For brevity, we shall follow Barrow by using the term '''inflection''' to mean either reflection or refraction.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;130,&amp;#8239;136,&amp;#8239;171&amp;#8239;(n.&amp;#8239;78), citing [[#barrow-1669|Barrow, 1669]], pp.&amp;#8239;10&amp;#8239;(§&amp;#8239;11),&amp;#8239;22,&amp;#8239;111.&lt;/ref&gt;{{efn|Not until 1675 was the term ''inflection'' hijacked for diffraction by [[w:Robert Hooke|Hooke]] and Newton; see [[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;92–3 &amp;amp; n.&amp;#8239;29.}}

If the image of an object-point in the inflecting surface is ''stigmatic'', it is the common point of intersection of all the inflected rays (for a real image), or of all the inflected ray-lines produced back through the surface (for a virtual image); in either case, it is a ''point of intersection of all lines of sight'' to the object-point via the surface (produced rectilinearly through the surface if necessary). Hence a ray incident along the cathetus, when &quot;inflected&quot; (and produced if necessary), passes through the same image-point. But that ray is ''undeviated'': it is transmitted without refraction or reflected back along itself, so that the &quot;inflected&quot; ray and the resulting line of sight remain on the cathetus. Thus the image-point lies at the intersection of the cathetus and any other line of sight (whether the image is real or virtual). Conversely, if the image-point lies at the intersection of the cathetus and the line of sight, then, if &quot;the&quot; image-point is to be consistent, all such lines of sight must intersect the cathetus at the same point, and therefore must intersect each other at that point, which is therefore a stigmatic image. In short:
:{{box|padding=1ex|The cathetus rule is equivalent to the proposition that ''the image of the object-point is stigmatic within the working aperture, which admits the cathetus''.}}
Notice that the derivation of this equivalence ''does not depend on any law of reflection or refraction'' except that a normally-incident ray is undeviated. Thus the equivalence, whatever its importance or lack thereof, may be rightly assigned a status that the ancients wrongly assigned to the cathetus rule itself: the status of being as fundamental as the laws of reflection and refraction.

In the case of the sagittal image formed by inflection at a surface axially symmetrical about the cathetus, the image is stigmatic within a working aperture consisting of two infinitesimal areas, one containing the foot of the cathetus and the other containing a circle with its axis on the cathetus.

The cathetus admitted by the working aperture may be notional provided that it is unambiguous, so that we cannot move the cathetus without moving the &quot;[[#active|active]]&quot; part of the surface. For example, while the conditions of Newton's &quot;Ax.&amp;#8239;{{serif|VI}}&quot; do not say that the working aperture admits the cathetus, they do say that the inflecting surface is plane or spherical, which implies that it can be uniquely produced (extended) so as to admit a unique undeviated ray—the &quot;notional&quot; cathetus—for a given object-point. And under these conditions, according to the &quot;axiom&quot;, the image is stigmatic &quot;either accurately or without any sensible Error.&quot;

So, after the cathetus rule has been reduced to a peculiarity of the sagittal image and dismissed from the elementary teaching of optics, a proposition implying wider conditions under which the rule holds, &quot;either accurately or without any sensible Error&quot;, is put up as ''axiomatic'' at the beginning of the introductory treatise by the highest authority on the subject!

In the statements and applications of the cathetus rule by ancient and medieval opticians, the assumption of stigmatism is always unrecognized and sometimes patently absurd. Alhacen's retention of the rule for cylindrical and conical mirrors may be consigned to the absurd category, except in cases of bilateral symmetry about the plane of reflection, for which the working aperture may be reduced to an infinitesimally narrow strip; in those cases the assumption of stigmatism may still be inexact, but is at least not absurd. In the unrecognized category, but ''almost'' recognized, are the cases which exploit the axial symmetry to claim that the image-point is on the cathetus although it is viewed from off the cathetus; this reasoning tacitly assumes that the image-point stays put as the line of sight moves off the cathetus, which is true if the various lines of sight have a common intersection. For example, Alhacen, having established that the image of the center of the eye in a convex spherical mirror is on the cathetus, extends the argument to another point on the eye, although that point is seen from off the cathetus;&lt;ref&gt;[[#smith-2006|Smith, 2006]], pp.&amp;#8239;396–7.&lt;/ref&gt; and Tacquet argues from the same symmetry that the image of a rod aligned with the cathetus is likewise aligned with the cathetus, although it is best seen from off the cathetus.&lt;ref&gt;[[#tacquet-1669|Tacquet, 1669]], p.&amp;#8239;222 (Prop.&amp;#8239;19).&lt;/ref&gt; Apparently the first writer to recognize the ''necessity'' of stigmatism is Benedetti, who, in his sixth letter to Vimercato (see [[#sixth|above]]), introduces the counterexample of the spherical burning mirror by saying &quot;I will prove to you that at no point can all the reflected rays meet each other.&quot;&lt;ref&gt;[[#benedetti-1585|Benedetti, 1585]], p.&amp;#8239;342.&lt;/ref&gt;

But in the useful range of cases that satisfy the conditions of Newton's &quot;Ax.&amp;#8239;{{serif|VI}}&quot;—&amp;#8239;that the surface is plane or spherical, and that the angles of incidence are not too large—ancient and medieval investigators should indeed have found the cathetus rule to be true &quot;either accurately or without any sensible Error.&quot; That range of cases also includes the following:

* When we look nearly vertically into still water, the departure of the image from the cathetus is imperceptible, as conceded by Kepler,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], p.&amp;#8239;89, end of Prop.&amp;#8239;19.&lt;/ref&gt; confirmed by Barrow,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;131–2, except that the diagram is upside-down for an air-water surface.&lt;/ref&gt; and implied in Newton's &quot;''Cas''.&amp;#8239;1.&quot;
* The same applies to looking nearly vertically ''out of''&amp;#8202; the water (also covered by &quot;''Cas''.&amp;#8239;1&quot;), as shown by Barrow, who further implies that the &quot;absolute&quot; image is the limit of the &quot;relative&quot; (tangential) image as the eye approaches the cathetus,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;132–4.&lt;/ref&gt; which he calls the &quot;axis&quot; or &quot;radiant axis&quot;.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;108.&lt;/ref&gt;{{efn|Not to be confused with what he calls the &quot;optical axis&quot;, which is synonymous with his &quot;principal ray&quot; and passes through the center of the eye ([[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;137,&amp;#8239;141,&amp;#8239;171&amp;#8239;n.&amp;#8239;79).}}
* Parallel incident rays refracted by a spherical surface, with small deviations, cut the axis at nearly the same point, as noted by Kepler,&lt;ref&gt;[[#kepler-donahue-00|Kepler/Donahue, 2000]], pp.&amp;#8239;205–6 (Prop.&amp;#8239;15).&lt;/ref&gt; and by Barrow,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;144.&lt;/ref&gt; who shows that an object-point at a finite distance gives the same result,&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;147.&lt;/ref&gt; in agreement with Newton's &quot;''Cas''.&amp;#8239;3.&quot;

These are some of the reasons why Newton's &quot;axiom&quot;, together with the case of the sagittal image, have been useful enough to launch the cathetus rule on a second, incognito career.

== A cathetus by any other name… ==

=== Anon. ===

[[File:Hyperbolic mirror.svg|thumb|300px|Stigmatic image {{math|''F''&lt;sub&gt;2&lt;/sub&gt;}} of a virtual object-point{{math| ''F''&lt;sub&gt;1&lt;/sub&gt;&amp;#8202;,}} formed by reflection in a convex hyperboloidal mirror with foci {{math|''F''&lt;sub&gt;1 &lt;/sub&gt;}}and{{math| ''F''&lt;sub&gt;2&lt;/sub&gt;&amp;#8202;}}. Rays initially directed toward{{math| ''F''&lt;sub&gt;1&lt;/sub&gt;}} are reflected through{{math| ''F''&lt;sub&gt;2&lt;/sub&gt;&amp;#8202;,}} including the undeviated ray{{math| ''F''&lt;sub&gt;2&amp;#8202;&lt;/sub&gt;''F''&lt;sub&gt;1&lt;/sub&gt;&amp;#8202;,}} which is the cathetus. Thus the image-point is the intersection of the cathetus with any other reflected ray. (Diagram by &amp;lsquo;Episcophagus&amp;rsquo; at ''Wikimedia Commons''.)]]

The equivalence between stigmatism and the cathetus rule is apparent in any diagram that shows a single surface bringing many rays from a single object-point to a focus at a single image-point, with one of the rays perpendicular to the surface. The (actual or assumed) stigmatism of the image is shown by the concurrence of the lines, and the point of concurrence is the point where every refracted or reflected ray (produced if necessary) meets the undeviated perpendicular ray—the cathetus. Such diagrams are offered in the widely-used text by Jenkins &amp;amp; White ([[#jenkins-white-76|1976]]) on pp.&amp;nbsp;47, 48, 49, and 100 (Fig.&amp;#8239;6B), the first and last being for an object at infinity. In each of these cases, the authors take the surface to be spherical (so that the stigmatism is only approximate) and the perpendicular ray is identified only by its passing through the center of curvature.

If the image of an object-point is stigmatic, it is uniquely located by ''any two rays'' belonging to that object-point, and we might as well choose those rays for convenience. For a single surface, the most obvious convenience is to let one of the rays be the one along the cathetus, so that it is undeviated. The location of the image then becomes a straightforward but unacknowledged application of the cathetus rule. This is how the image-point is located in our [[#Introduction:_Undeniable_implausibility|Figure&amp;#8239;1]] above. This is how Jenkins &amp;amp; White ([[#jenkins-white-76|1976]], pp.&amp;#8239;56–7) and [[w:George S. Monk|Monk]] ([[#monk-63|1963]], pp.&amp;#8239;8–9) derive the &quot;Gaussian formula&quot; relating the object and image distances for a spherical refracting surface—without explaining that the generality of the angles implies the stigmatism of the image within the accuracy of the formula.

=== Axis ===

The convenience of choosing a ray along the cathetus is multiplied if the object-point is on the axis of a system with several coaxial surfaces, so that the axis is perpendicular to all the surfaces. Then the image formed by the first surface is on the axis, which is therefore the cathetus for the second surface, which therefore forms another image on the axis, and so on, so that the axis serves the common cathetus for all the surfaces, and the final image is where the final refracted or reflected ray cuts that common cathetus. Thus Jenkins &amp;amp; White ([[#jenkins-white-76|1976]]) explain how to locate the image of an object-point on the axis of two thin lenses (pp.&amp;#8239;68–9, Fig.&amp;#8239;4{{serif|I}}), or of one thick lens (pp.&amp;#8239;78–9, Fig.&amp;#8239;5A),&lt;ref&gt;''Cf''. [[#hecht-17|Hecht, 2017]], p.&amp;#8239;167, Fig.&amp;#8239;5.14 (b) &amp;amp; (c).&lt;/ref&gt; especially for an object-point at infinity (pp.&amp;#8239;84–5, Fig.&amp;#8239;5G); the intermediate steps need not detain us (yet), except that their purpose is to find where the final refracted ray cuts the axis, because &quot;the axis itself is considered as the second light ray&quot; (p.&amp;#8239;69; ''cf''.&amp;nbsp;p.&amp;#8239;79). The beginnings of this approach may be discerned in [[w:Bonaventura Cavalieri|Bonaventura Cavalieri]]'s &quot;Six Geometrical Exercises&quot; of 1647.&lt;ref&gt;[[#cavalieri-1647|Cavalieri, 1647]], p.&amp;#8239;464ff; [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;127–8.&lt;/ref&gt;

But Barrow calls the cathetus the axis where there is only one surface, axially symmetrical about it.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;108.&lt;/ref&gt; Jenkins &amp;amp; White ([[#jenkins-white-76|1976]]) do likewise in diagrams showing the focal points of a spherical refracting surface (p.&amp;#8239;46; four cases)&lt;ref&gt;Cases (b) and (d) are respectively equivalent to Figs. 5.11 and 5.10 in Hecht ([[#hecht-17|2017]], p.&amp;#8239;165), except that Hecht does not name the &quot;axis&quot;.&lt;/ref&gt; and a spherical reflecting surface (p.&amp;#8239;99; two cases)&lt;ref&gt;Also in Fig.&amp;#8239;5.61 in Hecht ([[#hecht-17|2017]], p.&amp;#8239;196).&lt;/ref&gt;; and in those cases where the image is at a finite distance, its assumed stigmatism is seen from the concurrence of the ray-lines, and its location is seen to be consistent with the cathetus rule.

Jenkins &amp;amp; White ([[#jenkins-white-76|1976]], pp.&amp;#8239;56–7) and Monk ([[#monk-63|1963]], pp.&amp;#8239;8–9) even use the word ''axis'' in their derivations of the &quot;Gaussian formula&quot;, albeit only in the text. Here Jenkins &amp;amp; White make ad-hoc approximations from the outset, and Monk does so at the second step. For reasons which will become apparent, we shall now re-derive this formula in a more disciplined manner, introducing assumptions only as they are needed, after showing what can be deduced without them.&lt;span id=&quot;f-sagit&quot;&gt;&lt;/span&gt;

[[File:Refraction-at-spherical-surface.svg|thumb|720px|Distances and angles for refraction at a spherical surface. (Diagram by the author.)]]

Let {{mvar|O}} be an object-point facing a spherical refracting surface (separating two homogeneous isotropic media) whose radius of curvature is {{mvar|r}} (positive if convex as seen from{{mvar| O}}) with center{{mvar| C}}, so that {{mvar|OC}} is the cathetus ([[#f-sagit|Figure&amp;nbsp;6]]). Let {{mvar|V}} (for ''vertex'') be the foot of the cathetus, at a distance ''s'' from{{mvar| O}}.&amp;#8239; Let the point of refraction be{{mvar| P}}.&amp;#8239; The ''axial'' symmetry of the interface and media about the cathetus{{mvar| OC}}&amp;#8202; implies a bilateral symmetry about the plane of the cathetus and the incident ray{{mvar| OP}}, which in turn implies that the refracted ray must remain in that plane.{{efn|Alternatively we can argue that by the bilateral symmetry, the normal to the surface at{{mvar| P}}&amp;#8202; is in the plane of symmetry, which is therefore the plane of the incident ray and the normal, whence, by the law first articulated by Ptolemy, the refracted ray is in that plane. But I submit that the symmetry is enough, and that the law of Ptolemy follows from it.}} So let the point{{mvar| I}}, at a distance ''s&amp;prime;''&amp;#8202; from{{mvar| V}}, be the intersection of the refracted ray and the cathetus (if the refracted ray is parallel to the cathetus, we shall consider {{mvar|I}}&amp;#8202; to be at infinity). If angle {{mvar|OCP}} is called ''&amp;alpha;'', then, treating ''&amp;alpha;'' and ''ϕ'' (in [[#f-sagit|Figure&amp;nbsp;6]]) as exterior angles of triangles, we find that the remote interior angles at {{mvar|I}} and{{mvar| O}}&amp;#8202; are respectively ''&amp;alpha;&amp;minus;ϕ&amp;prime;'' and ''ϕ&amp;minus;&amp;alpha;'' (as labeled).

Now it is clear from the symmetry that ''s&amp;prime;'' is an ''even'' function of ''&amp;alpha;&amp;minus;ϕ&amp;prime;''. This, together with the smoothness of the function (apart from the [[w:Removable singularity|removable singularity]] at ''&amp;alpha;&amp;minus;ϕ&amp;prime; ''&amp;equals;&amp;#8239;0), implies that the graph of ''s&amp;prime;''&amp;#8202; vs. ''&amp;alpha;&amp;minus;ϕ&amp;prime;''&amp;#8202; passes through the ''s&amp;prime;'' axis with a slope of zero, so that the intersection {{mvar|I}}&amp;#8202; is stationary as the observation point (on{{mvar| PI}}, beyond{{mvar| I}}&amp;#8202;) passes through the cathetus{{mvar| OC}}. For the given object-point{{mvar| O}}, this stationarity of{{mvar| I}} is the limit of the intersection of a refracted ray with the cathetus as ''&amp;alpha;&amp;minus;ϕ&amp;prime;&amp;#8239;''&amp;rarr;&amp;#8239;0 (as claimed by Barrow), hence the limit of the intersection of two refracted rays with each other as both approach the cathetus, hence the limit of the tangential image-point as the observation point approaches the cathetus (as shown by Barrow). The limiting position of{{mvar| I}}, by construction, is on the cathetus, salvaging the cathetus rule as an approximation for small angles; and because the limit is a stationarity, the deviation from the limit, measured along the cathetus, is at worst 2nd-order in the angles (in which case the ray aberration is of 3rd order, as expected). This implies near-stigmatism for sufficiently small angles—justifying Newton's &quot;axiom&quot;.

All this has been shown from symmetry and smoothness, without relying on the exact law of refraction—or even the exact sphericity of the surface, provided that it is axially symmetric about the cathetus and sufficiently smooth. But now let us invoke the sphericity with center{{mvar| C}}, so that the segment{{mvar| CP}} (in [[#f-sagit|Figure&amp;nbsp;6]]) has length{{mvar| r}}. Let the distances {{mvar|OP}} and{{mvar| PI}}&amp;#8202; be respectively ''&amp;sigma;'' and ''&amp;sigma;&amp;prime;'' (as shown). Then, by the [[w:Law of sines|sine rule]] in triangle{{mvar| OCP}}, we have
::&lt;math&gt;\frac{r}{\,\sigma\,} = \frac{\sin(\phi-\alpha)}{\sin{\alpha}}&lt;/math&gt;
or, after expanding the sine of the difference and simplifying,
{{NumBlk|::|&lt;math&gt;
  \frac{r}{\,\sigma\,} = \sin\phi\,\cot\alpha - \cos\phi \,.
&lt;/math&gt;|{{EquationRef|1}}}}
Similarly, applying the sine rule in triangle{{mvar| ICP}} (and noting that the exterior angle has the same sine as its supplementary interior angle), we have
::&lt;math&gt;\frac{r}{\,\sigma'} = \frac{\sin(\alpha-\phi')}{\sin{\alpha}} \,,&lt;/math&gt;
i.e.
{{NumBlk|::|&lt;math&gt;
  \frac{r}{\,\sigma'} = \cos\phi' -\, \sin\phi'\cot\alpha \,.
&lt;/math&gt;|{{EquationRef|2}}}}
To eliminate ''&amp;alpha;'', we multiply ({{EquationNote|1}}) by &lt;math&gt;\tfrac{\sin\phi'}{r}&lt;/math&gt;,&amp;#8239; and ({{EquationNote|2}}) by &lt;math&gt;\tfrac{\sin\phi}{r}&lt;/math&gt;,&amp;#8239; and add the results, obtaining
{{NumBlk|::|&lt;math&gt;
  \frac{\sin\phi'}{\sigma} + \frac{\sin\phi}{\sigma'}
  \,=\, \frac{\sin\phi\,\cos\phi' -\, \cos\phi\,\sin\phi'}{r} \,.
&lt;/math&gt;|{{EquationRef|3}}}}

For the purpose of locating{{mvar| I}}, let us rearrange ({{EquationNote|3}}) as
{{NumBlk|::|&lt;math&gt;
  \frac{1}{\,\sigma'} \,=\, \frac{\sin(\phi-\phi')}{r\sin\phi}
                            - \frac{\sin\phi'}{\sigma\sin\phi} \,.
&lt;/math&gt;|{{EquationRef|4}}}}
Then, for paraxial rays, the angles ''ϕ'' and ''ϕ&amp;prime;'' are small so that the sines may be approximated by their arguments, and ''&amp;sigma;'' and ''&amp;sigma;&amp;prime;'' may be approximated by ''s'' and ''s&amp;prime;'' respectively, the fractional errors being 2nd-order in the angles. Thus we have
{{NumBlk|::|&lt;math&gt;
  \frac{1}{\,s'} \approx \frac{\,\phi-\phi'}{r\phi} - \frac{\,\phi'}{s\phi} \,.
&lt;/math&gt;|{{EquationRef|5}}}}
As {{mvar|CP}} is the radius of the spherical interface ([[#f-sagit|Figure&amp;nbsp;6]]), it is the normal to the interface at{{mvar| P}}, whence ''ϕ'' and ''ϕ&amp;prime;'' are the angles of incidence and refraction. Kepler did not know the exact law of refraction (although he had corresponded with Harriot, who did&lt;ref&gt;[[#lohne-59|Lohne, 1959]]; [[#shirley-51|Shirley, 1951]].&lt;/ref&gt;); but he was satisfied that for small angles, the ratios &lt;math&gt;\tfrac{\phi{-}\phi'}{\phi}&lt;/math&gt; and &lt;math&gt;\tfrac{\,\phi'}{\phi}&lt;/math&gt; are approximately constant,&lt;ref&gt;That he was aware of this fact as early as 1604 is shown in [[#kepler-donahue-00|Kepler/Donahue, 2000]], pp. 124, 127-9 (Prop.&amp;#8239;8), &amp;amp; 205–6 (Prop.&amp;#8239;15)—although he made greater use of it in his ''Dioptrice'' of 1611, where it is stated up-front as &quot;{{serif|VII}}. Axioma&quot; [&amp;zwj;[[#kepler-1859|Kepler (1859)]], p.&amp;#8239;529]. ''Cf''. [[#darrigol-12|Darrigol, 2012]], pp.&amp;#8239;34–5; [[#dijksterhuis-99|Dijksterhuis, 1999]], p.&amp;#8239;29; [[#malet-2003|Malet, 2003]], p.&amp;#8239;109; [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;126–7.&lt;/ref&gt; in which case, by ({{EquationNote|5}}), for given {{mvar|r}} and ''s'', the length ''s&amp;prime;''&amp;#8202; is approximately constant. The same conclusion applies to ''reflection'', for which we put ''ϕ&amp;prime; &amp;equals;&amp;#8239;&amp;minus;ϕ'' in ({{EquationNote|5}}) and write ''&amp;minus;s&amp;prime;''&amp;#8202; for ''s&amp;prime;''&amp;#8202; (that is, change the positive direction of ''s&amp;prime;''&amp;#8202;), obtaining
{{NumBlk|::|&lt;math&gt;
  \frac{\,1\,}{s} + \frac{1}{\,s'} \approx -\frac{2}{\,r\,} \,.
&lt;/math&gt;|{{EquationRef|6}}}}
Barrow first published this result.&lt;ref&gt;Expressed as an equation by Shapiro ([[#shapiro-1990|1990]], p.&amp;#8239;140), and matching [[#jenkins-white-76|Jenkins &amp;amp; White, 1976]], p.&amp;#8239;103, Eq.&amp;#8239;(6b). On the priority of Barrow (vs. Huygens), see Shapiro, p.&amp;#8239;128, and [[#dijksterhuis-99|Dijksterhuis, 1999]], pp.&amp;#8239;39,&amp;#8239;86.&lt;/ref&gt;

Having seen what can be done ''without'' the exact law of refraction, let us now invoke it: if {{mvar|n}} and{{mvar| n'}} denote the refractive indices of the two media ([[#f-sagit|Figure&amp;nbsp;6]]), then the ratio &lt;math&gt;\tfrac{n}{\sin\phi'}&lt;/math&gt; is the same as &lt;math&gt;\tfrac{n'}{\sin\phi}&lt;/math&gt;.&amp;#8201; Multiplying the exact equation ({{EquationNote|3}}) by this ratio, in the first form for terms in&amp;#8202; &lt;math&gt;\sin\phi'&lt;/math&gt; and the second for terms in&amp;#8202; &lt;math&gt;\sin\phi&lt;/math&gt;, we get
{{NumBlk|::|&lt;math&gt;
  \frac{n}{\,\sigma\,} + \frac{\,n'}{\,\sigma'}
  = \frac{n'\cos\phi' -\, n\cos\phi}{r} \,.
&lt;/math&gt;|{{EquationRef|7}}}}
For paracathetal&amp;#10744;paraxial rays, the cosines may be replaced by 1&amp;#8239; while ''&amp;sigma;''&amp;#8202; and ''&amp;sigma;&amp;prime;''&amp;#8202; may be replaced by ''s''&amp;#8202; and ''s&amp;prime;''&amp;#8202; (the fractional errors again being 2nd-order in the angles), to obtain
{{NumBlk|::|&lt;math&gt;
  \frac{n}{\,s\,} + \frac{\,n'}{\,s'} \approx \frac{n'\! - n}{r} \,,
&lt;/math&gt;|{{EquationRef|8}}}}
which is well known as the '''Gaussian formula''' for a spherical refracting surface,&lt;ref&gt;[[#jenkins-white-76|Jenkins &amp;amp; White, 1976]], pp.&amp;#8239;48,&amp;#8239;56.&lt;/ref&gt; although Barrow again gives an equivalent result.&lt;ref&gt;[[#shapiro-1990|Shapiro, 1990]], p.&amp;#8239;147 (for refractive indices 1 and{{mvar| n}}).&lt;/ref&gt; For ''reflection'', we put ''ϕ&amp;prime; &amp;equals;&amp;#8239;&amp;minus;ϕ''&amp;#8239; and {{mvar|n'&amp;#8239;&amp;equals;&amp;#8239;&amp;minus;n}}&amp;#8239; in ({{EquationNote|7}}) and ({{EquationNote|8}}) and change the positive directions of ''&amp;sigma;&amp;prime;'' and ''s&amp;prime;'', obtaining
{{NumBlk|::|&lt;math&gt;
  \frac{1}{\,\sigma\,} + \frac{1}{\,\sigma'} = -\frac{2\cos\phi}{r}
&lt;/math&gt;|{{EquationRef|9}}}}
for the exact result, and ({{EquationNote|6}}) again for the paracathetal&amp;#10744;paraxial approximation.

For reflection in a ''plane'' mirror, we put {{mvar|r&amp;#8239;&amp;rarr;&amp;#8239;&amp;infin;}}&amp;#8202; in the exact equation ({{EquationNote|9}}), which then reduces to ''&amp;sigma;&amp;prime; &amp;equals;&amp;#8239;&amp;minus;&amp;sigma;''&amp;#8202; for all ''ϕ'', confirming that the image is stigmatic, on the cathetus, and as far behind the mirror as the object-point is in front. Later we shall find other uses for the exact equations ({{EquationNote|7}}) and ({{EquationNote|9}}).

=== Auxiliary axis ===

From an object-point ''off''&amp;#8202; the axis of a coaxial system, a cathetus dropped to a facing spherical surface is not generally an axis of the whole system. But it is still an axis of that surface—wherefore it may be called an ''auxiliary axis'', while the axis of the system may be called the ''principal axis''—and a ray incident along that cathetus still offers the convenience of being undeviated by that surface. This convenience is exploited by Jenkins &amp;amp; White ([[#jenkins-white-76|1976]]) to find the image formed by refraction into a denser medium at a convex surface (p.&amp;#8239;51, Fig.&amp;#8239;3F) or a concave surface (p.&amp;#8239;52, Fig.&amp;#8239;3G), or by reflection at a concave surface (pp.&amp;#8239;100–101, Fig.&amp;#8239;6E) or a convex surface (p.&amp;#8239;101 &amp;amp; Fig.&amp;#8239;6F).&lt;ref&gt;The last two examples are also given by Hecht ([[#hecht-17|2017]], p.&amp;#8239;197, Fig.&amp;#8239;5.63), except that he does not use the term ''auxiliary axis'', but explains the concept using &quot;Ray-1&quot; in his Fig.&amp;#8239;5.62 (p.&amp;#8239;196).&lt;/ref&gt; In each case, one ray is chosen to pass through the center of curvature—that is, along the cathetus—and there are two candidates for a second ray, either of which (within the accuracy of the method) cuts the cathetus at the image-point.{{efn|The &quot;two candidates&quot;, one incident parallel to the principal axis and the other refracted parallel to that axis, would be enough by themselves, especially as the authors (Jenkins &amp;amp; White, [[#jenkins-white-76|1976]]) are describing what they call the ''parallel-ray method''; but, idiosyncratically, they mention the undeviated ray before the second parallel ray (p.&amp;#8239;51, and again on p.&amp;#8239;101).}}

For refraction at a single surface, as the same authors show (p.&amp;#8239;52 &amp;amp; Fig.&amp;#8239;3H), we can even use an auxiliary axis to locate the image of an object-point on the principal axis. First we construct the auxiliary axis parallel to the oblique incident ray from the object-point. This axis crosses the focal surface (which must be determined separately) at a point on the refracted oblique ray, fixing the direction of that ray, which then meets the principal axis at the desired image-point. In effect, the cathetus rule is used twice—first to find the image of a hypothetical object-point at infinity, fixing the direction of a refracted ray from the actual object-point, and second to find the image of that point on the cathetus from that point.{{efn|In the corresponding case for a concave mirror ([[#jenkins-white-76|Jenkins &amp;amp; White, 1976]], pp.&amp;#8239;101–2, Fig.&amp;#8239;6G), where the authors say &quot;If in place of ray 4 another ray were drawn through ''C'' and parallel to ray 3,&quot; they are referring to an auxiliary axis, but they do not actually draw it.}} The extension of the method to multiple surfaces is obvious.

The same authors, in a diagram already mentioned (p.&amp;#8239;48), show seven rays diverging from an object-point and refracted by a spherical surface to a real image-point, with one of the rays passing through the center of curvature but not otherwise labeled. In the corresponding diagram for reflection (p.&amp;#8239;100, Fig.&amp;#8239;6C), the ray through the center of curvature is labeled the auxiliary axis, and all the other rays are shown as cutting this ray at the image-point. In each case, the image as drawn (''assumed'' to be stigmatic) is located in accordance with the cathetus rule.

=== Undeviated ray ===

Wherever the cathetus rule holds—that is, wherever the image is stigmatic and the cathetus well defined—the necessary and sufficient property of the cathetus is that ''a ray incident along the cathetus is undeviated''. Thus, if the image of an object-point is approximately stigmatic within a working aperture that admits an approximately undeviated ray, then, subject to those approximations, the image lies at the intersection of the undeviated ray and any other emergent ray (produced if necessary) from the same object-point. In short, the approximately undeviated ray plays the role of the cathetus.

[[File:ThinLens.png|thumb|374px|Location of the image{{mvar| B&amp;prime;}} of an object-point{{mvar| B}}&amp;#8202; due to a thin lens. The (approximately) undeviated ray{{mvar| BOB&amp;prime;}} plays the role of the ancient cathetus: the image may be taken to be at the intersection of this ray and any other refracted ray originating at{{mvar| B}}. (Diagram by &amp;lsquo;Tamasflex&amp;rsquo; at ''Wikimedia Commons''.)]]

A ray through the center of a ''thin'' lens—that is, a lens whose thickness is negligible compared with the object and image distances—may be considered undeviated even if it is oblique to the principal axis. This ray plays the same role in Newton's &quot;''Cas''.&amp;#8239;4&quot; that the cathetus plays in his &quot;''Cas''.&amp;#8239;2&quot;.&lt;ref&gt;[[#newton-2010|Newton (2010)]], pp.&amp;#8239;11–13. More precisely, the nominated center is midway between the front and back focal points.&lt;/ref&gt; It plays the same role in Fig.&amp;#8239;4C of Jenkins &amp;amp; White ([[#jenkins-white-76|1976]], p.&amp;#8239;63) that the ray through the center of curvature plays in their Fig.&amp;#8239;3D (p.&amp;#8239;48), and (under the name &quot;chief ray&quot;) the same role in their Figs. 4B, 4D, &amp;amp; 4E (pp.&amp;#8239;62,&amp;#8239;63,&amp;#8239;64) that the cathetus respectively plays, anonymously in their Fig.&amp;#8239;3C (p.&amp;#8239;47) and as the &quot;auxiliary axis&quot; in their Figs. 3F &amp;amp; 3H (pp.&amp;#8239;51,&amp;#8239;53). More constructions reminiscent of the cathetus rule, with the ray through the center of the lens in the role of the cathetus, can be found in their Figs. 4F, 4G, 4H (for each lens), 4{{serif|I}} (ditto), and 7B, and in (e.g.) Figs. 5.23, 5.24, and 5.29 of Hecht ([[#hecht-17|2017]], pp.&amp;#8239;172,&amp;#8239;176).

For an object-point on the principal axis of the lens, the ray along that axis is ''exactly'' undeviated and serves as the cathetus for the entire lens, so that the cathetus rule applies to the entire lens if the image is stigmatic. Examples of this sort (again not mentioning the cathetus rule) can be discerned in Fig.&amp;#8239;4A of Jenkins &amp;amp; White, and in Fig. 5.15 of Hecht ([[#hecht-17|2017]], p.&amp;#8239;168).

== Off-axis astigmatism ==

The foregoing examples from Jenkins &amp;amp; White ([[#jenkins-white-76|1976]]) and Hecht ([[#hecht-17|2017]]) use [[w:Gaussian optics|Gaussian approximations]]. They can model [[w:Chromatic aberration|chromatic aberration]] if we allow for variation of refractive indices with wavelength. But if they are to model 3rd-order monochromatic aberrations in the [[w:Meridional ray|meridional]] plane ([[w:Spherical aberration|spherical aberration]], tangential [[w:Coma (optics)|coma]], curvature of the tangential focal surface, and [[w:Distortion (optics)|distortion]]), they must be modified—perhaps by resorting to trigonometric ray-tracing in the meridional plane,&lt;ref&gt;See, e.g., [[#born-wolf-02|Born &amp;amp; Wolf, 2002]], pp.&amp;#8239;204–7.&lt;/ref&gt; in which case we still have the problem of assessing aberrations that involve rays outside that plane. For sagittal coma we can use the well-known proportionality (to leading order) between sagittal and tangential coma.&lt;ref&gt;[[#jenkins-white-76|Jenkins &amp;amp; White, 1976]], p.&amp;#8239;164.&lt;/ref&gt; For [[w:Astigmatism (optical systems)|astigmatism]], however, we need a sample of rays outside the meridional plane. With spherical surfaces, the easiest way to take such a sample is to exploit the exactness of the cathetus rule for the sagittal image formed by a surface axially symmetrical about the cathetus. And this is where we reap the reward for delaying approximations in the above derivation of the &quot;Gaussian formula&quot;.

In our [[#f-sagit|Figure&amp;nbsp;6]], suppose that the line {{mvar|OC}} is ''not'' the principal axis, but only an auxiliary axis. Let{{mvar| O}} be an off-axis object-point or an intermediate image thereof; and from{{mvar| O}}, let {{mvar|OPI}}&amp;#8202; be the path of the '''chief ray'''—that is, the ray through the center of the main aperture (wherever the main aperture stop happens to be). Then the sagittal image formed by the surface{{mvar| VP}}&amp;#8202; is{{mvar| I}}, whose position is given by equation ({{EquationNote|7}}) for a refractive surface, or ({{EquationNote|9}}) for a reflective surface. Equivalent results are given by Jenkins &amp;amp; White, citing the derivation by Monk,&lt;ref&gt;[[#jenkins-white-76|Jenkins &amp;amp; White, 1976]], p.&amp;#8239;169, Eqs.&amp;#8239;(9p), 2nd eq. (for refraction) and p.&amp;#8239;111, 2nd eq. (for reflection), citing [[#monk-63|Monk, 1963]], pp.&amp;#8239;424–6.&lt;/ref&gt; who begins by saying that &quot;if coma is absent, all the rays which have the same inclination… as{{mvar| OP}} with{{mvar| OC}} will intersect the line{{mvar| OC}}… in a point&quot; which we call{{mvar| I}}. The condition that &quot;coma is absent&quot; is redundant because the conclusion follows from the axial symmetry about{{mvar| OC}} (which Monk ignores, calling {{mvar|PC}}&amp;#8202; the axis). No such condition is assumed in the earlier derivation by [[w:Alexander Eugen Conrady|Conrady]], first published in 1929,&lt;ref&gt;[[#conrady-92|Conrady (1992)]], pp.&amp;#8239;409–10.&lt;/ref&gt; which duly invokes the auxiliary axis, and which, in spite of its different sign convention, is the main source for our derivation of ({{EquationNote|7}}) above. Conrady's equation (d) corresponds to our ({{EquationNote|7}}), and agrees with the result that [[w:Principles of Optics|Born &amp;amp; Wolf]] obtain by a longer process, involving a &quot;thin pencil&quot; of rays and a Hamiltonian characteristic function.&lt;ref&gt;[[#born-wolf-02|Born &amp;amp; Wolf, 2002]], p.&amp;#8239;186, Eq.&amp;#8239;(22).&lt;/ref&gt; None of these sources uses the word ''cathetus'' or refers to the cathetus rule.

Corresponding expressions for the distance of the ''tangential'' image along{{mvar| PI}}&amp;#8202; are given by the same authors and—most remarkably—by Barrow.&lt;ref&gt;On Barrow, and Newton's deference to him in this matter, see [[#shapiro-1990|Shapiro, 1990]], pp.&amp;#8239;135–6,&amp;#8239;147–8, and [[#newton-anon-1728|Newton/anon., 1728]], p.&amp;#8239;107.&lt;/ref&gt; In principle, we locate the tangential image by moving{{mvar| P}} along the arc{{mvar| VP}} (by an infinitesimal distance if we want an analytical result, or a finite distance if we are tracing rays numerically) and finding the intersection of the new{{mvar| PI}} with the old. The distance between the tangential and sagittal images along the old{{mvar| PI}}&amp;#8202; is a measure of the astigmatism.

By the axial symmetry, as we scan the aperture by rotating the arc{{mvar| VP}} about the axis{{mvar| OC}}, the tangential image likewise rotates about that axis, tracing a circular arc; and as we scan the aperture by moving{{mvar| P}} away from{{mvar| V}}, the sagittal image can only move along that axis. So the tangential and sagittal focal lines are perpendicular to each other; but the sagittal focal line is ''not'' generally perpendicular to the chief ray{{mvar| PI}} (although the tangential focal line is). Thus, as Born &amp;amp; Wolf note, it is not generally true that the focal lines are perpendicular to the chief ray &quot;as is often incorrectly asserted in the literature&quot;.&lt;ref&gt;[[#born-wolf-02|Born &amp;amp; Wolf, 2002]], p.&amp;#8239;182. Earlier on the same page, Born &amp;amp; Wolf themselves may seem to have asserted what they now deny. But the exculpatory words are &quot;To the first order&quot;; for a ''thin'' pencil, if the distance between the focal lines measured along the central ray is first-order, then the obliquity of either focal line to the central ray is ''second-order''.&lt;/ref&gt; Indeed I have noticed that the offenders include Jenkins &amp;amp; White ([[#jenkins-white-76|1976]]), who claim that the sagittal focal line, which they call ''S'', is perpendicular to what they call the ''sagittal plane'' (p.&amp;#8239;169), which contains the chief ray and is perpendicular to the ''tangential plane'' (meridional plane; see their Fig.&amp;#8239;9P). They go on to say that on the sagittal focal surface, the images are &quot;parallel to the spokes&quot; (p.&amp;#8239;169), whereas in fact the sagittal focal line for a point on a spoke need only be in the plane of the spoke and the axis. Their Fig.&amp;#8239;6N (p.&amp;#8239;112) is similarly misleading; the sagittal focal line ''S'' should be along the auxiliary axis—that is, parallel to the incoming rays (the object-point being at infinity).

In our [[#f-sagit|Figure&amp;nbsp;6]], the ''sagittal plane'' after refraction is the plane perpendicular to the plane of the diagram and containing the ray{{mvar| PI}}. If we leave the sagittal plane fixed and rotate the point of refraction about the axis (cathetus){{mvar| OC}}, the circle traced on the refracting surface is not identical to the intersection of that surface with the sagittal plane, but is tangential to that intersection, and the tangency is enough for calculating the astigmatism to leading order.&lt;ref&gt;Compare the corresponding remarks by Conrady ([[#conrady-92|1992]], top of p.&amp;#8239;410).&lt;/ref&gt; Thus Born &amp;amp; Wolf get the same sagittal equation as Conrady in spite of their radically different method. In a coaxial system, as {{mvar|P}} traces a circle with axis{{mvar| OC}}, the path traced by the intersection of the refracted ray{{mvar| PI}} with the ''next'' surface is not generally a circle with its axis on the cathetus from {{mvar|I}}&amp;#8202; to that surface, but again is tangential to such a circle. Hence equation ({{EquationNote|7}}) or ({{EquationNote|9}}) can be used with successive surfaces to find the successive positions of the sagittal image on the chief ray, and assess the final astigmatism, to leading order.

== Conclusion: Unreasonable in what sense? ==

It has been shown that there are conditions under which the cathetus rule is true or nearly so. Let it be conceded that under these conditions the rule must be, in some sense, effective, and that this effectiveness, as far as it goes, is by definition reasonable. One might object that these conditions—that the image is stigmatic or nearly so, and the cathetus unambiguous—seem narrow, and that the effectiveness of the rule, by comparison, seems unreasonably wide. In response, one could point out that surfaces forming stigmatic or nearly stigmatic images are useful and therefore likely to be encountered in practice, and likely to encourage propagation of any principle found applicable to them. Moreover, the shapes nominated by Newton as producing nearly stigmatic images—plane or spherical, or, let us add, nearly so—may exist for reasons unrelated to their imaging properties: I may see my face reflected in a teapot, though the teapot is not an optical device. For these reasons, examples of the effectiveness of the rule might reasonably be prevalent, or at least prominent.

When we delve into the history of that &quot;antient and common&quot; principle, however, any semblance of reasonableness evaporates.

The cathetus rule was unanimously upheld for nearly 19 centuries although there was not a single non-tautological case in which the rule had been validly demonstrated. Even the tautological case—that in which the line of sight is along the cathetus—was botched from the beginning (recall Euclid's &quot;postulates&quot;), and eventually put on a secure footing after 13 centuries when Alhacen posed the examples of the eye lining up a sharp tip with its reflection, and the eye looking at its own reflection.

But, after Kepler's attack in 1604 sent the rule into decline, only one more century passed before the rule was rehabilitated, without acknowledgment, by Newton's widely applicable &quot;axiom&quot; of approximate stigmatism, whereby the cathetus—disguised as the axis or the auxiliary axis or (generalized) as the undeviated ray—made itself extremely useful in &quot;Gaussian&quot; optics. Meanwhile the ''exact'' application of the rule to the sagittal image, for axial symmetry about the cathetus, languished in Newton's posthumous lecture notes, but reappeared in the 20th century—unnamed and unsourced—for the evaluation of 3rd-order astigmatism in coaxial systems with spherical surfaces, yielding the same formula as Hamiltonian theory, with less labor and less conceptual difficulty.

For nearly nineteen centuries, until Benedetti (1585), the cathetus rule was a non-sequitur: the effectiveness of the rule, in so far as it was correctly described, was unreasonably unexplained. For the three centuries since Newton, it has been unreasonably unrecognized.

== Additional information ==

=== Acknowledgments ===

If my analysis of Benedetti ([[#benedetti-1585|1585]]) adds any value to Goulding's ([[#goulding-18|2018]]), much of the credit is due to [[w:Google Translate|Google Translate]] and [[w:ChatGPT|ChatGPT]] 3.5; the latter (with a few &quot;custom instructions&quot;) expedited the correction of [[w:Optical character recognition|OCR]] errors in the plain text from [[w:Google Books|Google Books]], and then gave a second opinion on translation.

=== Competing interests ===
None.

=== Ethics statement ===
This article does not concern research on human or animal subjects.

== Notes ==

{{notelist|25em}}

== References ==

{{reflist|16em}}

== Bibliography ==

&lt;div style=&quot;font-size: 111%&quot;&gt;

{{refbegin|indent=yes}}

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*&lt;span id=&quot;schuster-00&quot;&gt;J.A. Schuster, 2000, &quot;Descartes ''opticien'': The construction of the law of refraction and the manufacture of its physical rationales, 1618–29&quot;, in S.&amp;#8239;Gaukroger, J.A.&amp;#8239;Schuster, &amp;amp; J.&amp;#8239;Sutton (eds.), ''Descartes' Natural Philosophy'', London: Routledge, pp.&amp;#8239;258–312.&lt;/span&gt;
*&lt;span id=&quot;shapiro-1990&quot;&gt;A.E. Shapiro, 1990, &quot;The ''Optical Lectures'' and the foundations of the theory of optical imagery&quot;, in M.&amp;#8239;Feingold (ed.), ''Before Newton: The Life and Times of Isaac Barrow'', Cambridge, pp.&amp;#8239;105–78.&lt;/span&gt;
*&lt;span id=&quot;shapiro-2008&quot;&gt;A.E. Shapiro, 2008, &quot;Images: Real and Virtual, Projected and Perceived, from Kepler to Dechales&quot;, ''Early Science and Medicine'', vol.&amp;#8239;13, no.&amp;#8239;3, pp.&amp;#8239;270–312; [https://www.jstor.org/stable/20617731 jstor.org/stable/20617731].&lt;/span&gt;
*&lt;span id=&quot;shirley-51&quot;&gt;J.W. Shirley, 1951, &quot;An early experimental determination of Snell's law&quot;, ''American Journal of Physics'', vol.&amp;#8239;19, no.&amp;#8239;9 (Dec.&amp;#8239;1951), pp.&amp;#8239;507–8; [https://doi.org/10.1119/1.1933068 doi.org/10.1119/1.1933068].&lt;/span&gt;
*&lt;span id=&quot;smith-1996&quot;&gt;A.M. Smith (tr.), 1996, &quot;Ptolemy's theory of visual perception: An English translation of the ''Optics'' with introduction and commentary&quot;, ''Transactions of the American Philosophical Society'', vol.&amp;#8239;86, no.&amp;#8239;2; [https://www.jstor.org/stable/3231951 jstor.org/stable/3231951].&lt;/span&gt;
*&lt;span id=&quot;smith-2001&quot;&gt;A.M. Smith (tr.&amp;#8239;&amp;amp;&amp;#8239;ed.), 2001, &quot;Alhacen's theory of visual perception: A critical edition, with English translation and commentary, of the first three books of Alhacen's ''De&amp;nbsp;Aspectibus'', the medieval Latin version of Ibn&amp;#8239;al-Haytham's ''Kitāb al-Manāzir''&quot;, in ''Transactions of the American Philosophical Society'', vol.&amp;#8239;91, no.&amp;#8239;4, [https://www.jstor.org/stable/3657358 jstor.org/stable/3657358] (vol.&amp;#8239;1: Introduction and Latin text), and no.&amp;#8239;5, [https://www.jstor.org/stable/3657357 jstor.org/stable/3657357] (vol.&amp;#8239;2: English translation).&lt;/span&gt;
*&lt;span id=&quot;smith-2006&quot;&gt;A.M. Smith (tr.&amp;#8239;&amp;amp;&amp;#8239;ed.), 2006, &quot;Alhacen on the principles of reflection: A critical edition, with English translation and commentary, of Books 4 and 5 of Alhacen's ''De&amp;nbsp;Aspectibus'', the medieval Latin version of Ibn&amp;#8239;al-Haytham's ''Kitāb al-Manāzir''&quot;, in ''Transactions of the American Philosophical Society'', vol.&amp;#8239;96, no.&amp;#8239;2, [https://www.jstor.org/stable/20020399 jstor.org/stable/20020399] (vol.&amp;#8239;1: Introduction and Latin text), and no.&amp;#8239;3, [https://www.jstor.org/stable/20020403 jstor.org/stable/20020403] (vol.&amp;#8239;2: English translation).&lt;/span&gt;
*&lt;span id=&quot;smith-2008&quot;&gt;A.M. Smith (tr.&amp;#8239;&amp;amp;&amp;#8239;ed.), 2008, &quot;Alhacen on image-formation and distortion in mirrors: A critical edition, with English translation and commentary, of Book 6 of Alhacen's ''De&amp;nbsp;Aspectibus'', the medieval Latin version of Ibn&amp;#8239;al-Haytham's ''Kitāb al-Manāzir''&quot; (vol.&amp;#8239;2: English translation), ''Transactions of the American Philosophical Society'', vol.&amp;#8239;98, no.&amp;#8239;1, sec.&amp;#8239;2; [https://www.jstor.org/stable/27757399 jstor.org/stable/27757399].&lt;/span&gt;
*&lt;span id=&quot;smith-2010&quot;&gt;A.M. Smith (tr.&amp;#8239;&amp;amp;&amp;#8239;ed.), 2010, &quot;Alhacen on Refraction: A critical edition, with English translation and commentary, of Book 7 of Alhacen's ''De&amp;nbsp;Aspectibus'', the medieval Latin version of Ibn&amp;#8239;al-Haytham's ''Kitāb al-Manāzir''&quot; (vol.&amp;#8239;2: English translation), ''Transactions of the American Philosophical Society'', vol.&amp;#8239;100, no.&amp;#8239;3, sec.&amp;#8239;2; [https://www.jstor.org/stable/20787651 jstor.org/stable/20787651].&lt;/span&gt;
*&lt;span id=&quot;smith-2017&quot;&gt;A.M. Smith, 2017, ''From Sight to Light: The Passage from Ancient to Modern Optics'', University of Chicago Press, 2015 (paperback ed., 2017).&lt;/span&gt;
*&lt;span id=&quot;tacquet-1669&quot;&gt;A.&amp;#8239;Tacquet, 1669, ''Catoptrica Tribus Libris Exposita'', in ''Opera Mathematica'', Antwerp: Meursius, vol.&amp;#8239;2, [https://books.google.com/books?id=XHK2NgG3UfQC google.com/books?id=XHK2NgG3UfQC], pp.&amp;#8239;213–264ff.&lt;/span&gt;
*&lt;span id=&quot;takahashi-92&quot;&gt;K.&amp;#8239;Takahashi, 1992, ''The Medieval Latin Traditions of Euclid's''&amp;#8202; Catoptrica: ''A Critical Edition of''&amp;#8201; De speculis ''with an Introduction, English Translation and Commentary'', Kyushu University Press.&lt;/span&gt;
*&lt;span id=&quot;unguru-72&quot;&gt;S. Unguru, 1972, &quot;Witelo and thirteenth-century mathematics: An assessment of his contributions&quot;, ''Isis'', vol.&amp;#8239;63, no.&amp;#8239;4 (Dec.&amp;#8239;1972), pp.&amp;#8239;496–508; [https://www.jstor.org/stable/229773 jstor.org/stable/229773].&lt;/span&gt;
*&lt;span id=&quot;vanHelden-et-al-10&quot;&gt;A.&amp;#8239;Van Helden, S.&amp;#8239;Dupré, R.&amp;#8239;van Gent, &amp;amp; H.&amp;#8239;Zuidervaart (eds.), 2010, ''The Origins of the Telescope'', Amsterdam: KNAW Press; [https://dspace.library.uu.nl/handle/1874/224188 dspace.library.uu.nl/handle/1874/224188] (open access).&lt;/span&gt;
*&lt;span id=&quot;vollgraff-1936&quot;&gt;J.A.&amp;#8239;Vollgraff, 1936, &quot;Snellius' notes on the reflection and refraction of rays&quot;, ''Osiris'', vol.&amp;#8239;1 (Jan.&amp;#8239;1936), pp.&amp;#8239;718–25; [https://www.jstor.org/stable/301634 jstor.org/stable/301634].&lt;/span&gt;

{{refend}}

&lt;/div&gt;

== Further reading ==

Of all the authors cited above, Goulding ([[#goulding-18|2018]]), although his account ends early in the 17th century, has by far the most information on the cathetus rule (including much detail on the modifications by Brengger and Stevin), and he alone reports Benedetti's priority in disproving and salvaging the rule.

For a concise general history of optics over the life of the traditional cathetus rule, see A.&amp;#8239;Mark Smith, &quot;Optics to the time of Kepler&quot;, ''Encyclopedia of the History of Science'' (Nov.&amp;#8239;2022; rev. Jul.&amp;#8239;2023), [https://doi.org/10.34758/v9kd-ad56 doi.org/10.34758/v9kd-ad56]. For a more expansive version, see [[#smith-2017|Smith, 2017]].

[[Category:Physics]]
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      <text bytes="6834" sha1="7btaxcy67ovwzjtsoswhlrm64prtrzl" xml:space="preserve">&lt;noinclude&gt;{{title|Using generative AI guidelines}}
__TOC__

&lt;noinclude&gt;==In a nutshell==&lt;/noinclude&gt;
Generative artificial intelligence (genAI) use is permitted for the [[Motivation and emotion/Assessment/Major project|major project]], but not the [[Motivation and emotion/Assessment/Exam|exam]]. Acknowledge use of genAI in the [[Wikiversity:FAQ/Editing/Edit summary|
edit summary]], with a link to the conversation or the tool used and prompt details. Fact-check genAI content and only cite peer-reviewed sources which you have consulted. Human-rewrite genAI content to enhance quality.&lt;includeonly&gt; [[Motivation and emotion/Assessment/Using generative AI|More detail ...]]&lt;/includeonly&gt;&lt;noinclude&gt;

==Summary==
[[File:Deeply engrossed in puzzle.png|thumb|220x220px|'''Figure 1'''. &lt;!-- An image of an elderly woman deeply engrossed in her daily crossword puzzle. --&gt;This image was generated by [[Motivation and emotion|Motivation and Emotion]] student [[User:JorjaFive|JorjaFive]] using [[w:Midjourney|Midjourney]] and uploaded to [[commons:|Wikimedia Commons]] for use in the [[Motivation and emotion/Book/2023/Flourishing in the elderly|flourishing in the elderly]] chapter.]]

GenAI tools can help brainstorm, explain concepts, develop a structure, synthesise ideas, and improve the quality of written expression. GenAI tools can aid but should not replace independent thinking and reading of primary sources.

Acknowledge genAI use in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]]. Academia is based on transparency. Follow the principle that &quot;''more acknowledgment is better than less''&quot;. However, acknowledgement is not required for low-level tasks such as improving spelling and grammar.

You are responsible for content you submit. Be aware of limitations of genAI tools such as inaccuracies, biases, and incomplete content. Fact-check all claims and only cite peer-reviewed citations which you consulted.

The best results are obtained from genAI tools through carefully crafted prompting based on reading of primary sources. 

If you are unsure, post to [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Detailed guidelines==
;Use ethically, with caution
Learning to use genAI tools (such as [[w:ChatGPT|ChatGPT]], [[w:Claude (language model)|Claude]],
[[w:Gemini (chatbot)|Gemini]], and [[w:Microsoft Copilot|Microsoft Copilot]]) responsibly and ethically is an emerging skill. GenAI tools can be used to enhance academic work, but should be used judiciously and as a supplementary tool, rather than as a replacement for independent thinking and academic inquiry.

GenAI tools may be used to assist in preparation of the major project ([[Motivation and emotion/Assessment/Topic|topic development]] and [[Motivation and emotion/Assessment/Chapter|book chapter]]). 

;How to acknowledge
[[File:Wikipedia Edit Summary dialog in VisualEditor.png|thumb|450px|'''Figure 2'''. If contributing genAI content, include the tool and prompt details in the edit summary, with a link to the conversation]]

[[File:Edit summary for genAI content.png|thumb|450px|'''Figure 3'''. Example page history which demonstrates best practice edit summaries for contributing and revising genAI content]]

Use of GenAI tools must be clearly acknowledged in [[Wikiversity:FAQ/Editing/Edit summary|Wikiversity edit summaries]] (e.g., see Figure 2), otherwise it is a violation of academic integrity.

Best practice is to include a publicly accessible link to the chatbot conversation (e.g., [https://help.openai.com/en/articles/7925741-chatgpt-shared-links-faq ChatGPT shared links FAQ]). If a link can't be shared, then provide details about the tool and the prompt in the [[Wikiversity:FAQ/Editing/Edit summary|edit summary]], (e.g., &quot;ChatGPT May 24 2025 Version. ''Prompt detail or summary''&quot;) (see Figure 3). The chatbot conversation should ''not'' be included as a citation and listed in the references because it is not a reliable, primary, peer-reviewed source.

These practices help to ensure that the use of genAI is clear. Transparency is key to good practice in academia. If in doubt, err on the side of providing too much acknowledgement rather than not enough. However, there is no need to acknowledge genAI use for low-level tasks such as fixing grammar and spelling.

;Limitations
Be aware of the limitations of genAI tools. Content they generate may be inaccurate, biased, incomplete, or otherwise problematic. Minimal effort prompts yield low quality results. Refine prompts to get better outcomes. You are entirely responsible for the accuracy and quality of any content you submit.

;Fact-check and cite
Always fact check. Regardless of whether genAI has been used, all claims need to be supported by verified peer-reviewed citations which you have consulted. Low-energy or unreflective reuse of text generated by genAI large language models without further investigation and reviewing of primary, peer-reviewed academic literature is likely to lead to a poor quality result. GenAI tools work best for topics which you already understand. Guide and craft genAI responses based on your reading of peer-reviewed theory and research.

;Going forth
Despite these warnings, you are encouraged to explore use of genAI tools to help develop higher quality work. Recommended uses of genAI tools include:
* brainstorming
* explaining key concepts
* developing a structure
* synthesising complex ideas
* rephrasing to improve [[Motivation and emotion/Assessment/Chapter/Readability|readability]] and the quality of written expression
* checking spelling and grammar
* image generation (e.g., see Figure 1)
* scenario generation
* critical feedback and suggestions for improvement

If you are unsure about how to use genAI effectively or how to acknowledge its use appropriately, ask in [[Motivation and emotion/About/Discussion|discussions]], so we can all learn together.

==Example==
* [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Affiliation_motivation_across_cultures&amp;action=history Affiliation and motivation] (Book chapter, 2025)

==Learn about genAI==
* [https://canberra.libguides.com/genai GenAI for students] (University of Canberra Library)
* [https://techcrunch.com/2024/06/01/what-is-ai-how-does-ai-work/ WTF is AI?] provides a useful introduction and non-technical overview about how genAI works, what it is capable of, limitations, and issues

==See also==
* [[b:Wikibooks:Artificial Intelligence|Wikibooks:Artificial Intelligence]] (policy)
* [[w:Wikipedia:Large language models|Wikipedia:Large language models]] (information page)
* [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence]] (policy)

[[Category:Motivation and emotion/Assessment]]
[[Category:Generative artificial intelligence]]
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    <title>Talk:Motivation and emotion/Assessment/Using generative AI</title>
    <ns>1</ns>
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      <text bytes="2526" sha1="dhhfxyhiuxnbeqz358t4far54pijc2m" xml:space="preserve">==Changes for 2024==
The original 2023 guidelines have been revised in 2024 to make it clearer that any genAI-based text must:
* clearly and transparently provide genAI acknowledgement in the edit summary, with a link to the chatbot conversation or model and prompt details
* be fact-checked
* include appropriate peer-reviewed citation

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:24, 5 July 2024 (UTC)

==Literature searching==
I find this page useful but sometimes, I still prefer to find my sources through books and medical journals rather than use the help of Gen AI. [[User:Eva U3259916|Eva U3259916]] ([[User talk:Eva U3259916|discuss]] • [[Special:Contributions/Eva U3259916|contribs]]) 05:53, 10 August 2025 (UTC)

:{{ping|Eva U3259916}} For a thorough literature search, prioritise search of academic journal databases. Google Scholar provides very useful search. GenAI may be helpful for cross-checking and seeing whether it suggests any key literature that was missed during database searching. However, I would caution against relying on genAI as the primary tool for academic literature searching. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:39, 4 October 2025 (UTC)

== Proposal for rewriting this article  ==

The following is my proposed draft In its entirety: '''“Don’t.”''' [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 20:29, 29 April 2026 (UTC)
: Why? &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;font style=&quot;color:#f90;&quot;&gt;+&lt;/font&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &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;font style=&quot;color:#f90;&quot;&gt;+&lt;/font&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis</text>
      <sha1>dhhfxyhiuxnbeqz358t4far54pijc2m</sha1>
    </revision>
    <revision>
      <id>2816358</id>
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      <contributor>
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        <id>3054149</id>
      </contributor>
      <comment>/* Proposal for rewriting this article */ Reply</comment>
      <origin>2816358</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2883" sha1="70smpoe9cn7fudyo1cx47i75e695rwi" xml:space="preserve">==Changes for 2024==
The original 2023 guidelines have been revised in 2024 to make it clearer that any genAI-based text must:
* clearly and transparently provide genAI acknowledgement in the edit summary, with a link to the chatbot conversation or model and prompt details
* be fact-checked
* include appropriate peer-reviewed citation

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:24, 5 July 2024 (UTC)

==Literature searching==
I find this page useful but sometimes, I still prefer to find my sources through books and medical journals rather than use the help of Gen AI. [[User:Eva U3259916|Eva U3259916]] ([[User talk:Eva U3259916|discuss]] • [[Special:Contributions/Eva U3259916|contribs]]) 05:53, 10 August 2025 (UTC)

:{{ping|Eva U3259916}} For a thorough literature search, prioritise search of academic journal databases. Google Scholar provides very useful search. GenAI may be helpful for cross-checking and seeing whether it suggests any key literature that was missed during database searching. However, I would caution against relying on genAI as the primary tool for academic literature searching. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:39, 4 October 2025 (UTC)

== Proposal for rewriting this article  ==

The following is my proposed draft In its entirety: '''“Don’t.”''' [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 20:29, 29 April 2026 (UTC)
: Why? &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;font style=&quot;color:#f90;&quot;&gt;+&lt;/font&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC)
::I don’t need to explain why generative AI is bad. It’s already been done many, many times. [[Wikipedia|Our sister project]] is a good place to start. My “draft” is hyperbole, but it gets the point across. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 00:14, 21 June 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &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;font style=&quot;color:#f90;&quot;&gt;+&lt;/font&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis</text>
      <sha1>70smpoe9cn7fudyo1cx47i75e695rwi</sha1>
    </revision>
    <revision>
      <id>2816380</id>
      <parentid>2816358</parentid>
      <timestamp>2026-06-21T08:32:10Z</timestamp>
      <contributor>
        <username>~2026-35992-67</username>
        <id>3096066</id>
      </contributor>
      <origin>2816380</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2886" sha1="0yuq8a4wrswh8qrmlyq6t1c2xcmuxjy" xml:space="preserve">==Changes for 2024==
The original 2023 guidelines have been revised in 2024 to make it clearer that any genAI-based text must:
* clearly and transparently provide genAI acknowledgement in the edit summary, with a link to the chatbot conversation or model and prompt details
* be fact-checked
* include appropriate peer-reviewed citation

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:24, 5 July 2024 (UTC)

==Literature searching==
I find this page useful but sometimes, I still prefer to find my sources through books and medical journals rather than use the help of Gen AI. [[User:Eva U3259916|Eva U3259916]] ([[User talk:Eva U3259916|discuss]] • [[Special:Contributions/Eva U3259916|contribs]]) 05:53, 10 August 2025 (UTC)

:{{ping|Eva U3259916}} For a thorough literature search, prioritise search of academic journal databases. Google Scholar provides very useful search. GenAI may be helpful for cross-checking and seeing whether it suggests any key literature that was missed during database searching. However, I would caution against relying on genAI as the primary tool for academic literature searching. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:39, 4 October 2025 (UTC)

== Proposal for rewriting this article  ==

The following is my proposed draft In its entirety: '''“Don’t.”''' [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 20:29, 29 April 2026 (UTC)
: Why? &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; 23:12, 20 June 2026 (UTC)
::I don’t need to explain why generative AI is bad. It’s already been done many, many times. [[Wikipedia|Our sister project]] is a good place to start. My “draft” is hyperbole, but it gets the point across. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 00:14, 21 June 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &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;font style=&quot;color:#f90;&quot;&gt;+&lt;/font&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis</text>
      <sha1>0yuq8a4wrswh8qrmlyq6t1c2xcmuxjy</sha1>
    </revision>
    <revision>
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      <contributor>
        <username>~2026-35992-67</username>
        <id>3096066</id>
      </contributor>
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      <text bytes="2889" sha1="qyyrp0byfzqk4s2f0dftxj84epcjx62" xml:space="preserve">==Changes for 2024==
The original 2023 guidelines have been revised in 2024 to make it clearer that any genAI-based text must:
* clearly and transparently provide genAI acknowledgement in the edit summary, with a link to the chatbot conversation or model and prompt details
* be fact-checked
* include appropriate peer-reviewed citation

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:24, 5 July 2024 (UTC)

==Literature searching==
I find this page useful but sometimes, I still prefer to find my sources through books and medical journals rather than use the help of Gen AI. [[User:Eva U3259916|Eva U3259916]] ([[User talk:Eva U3259916|discuss]] • [[Special:Contributions/Eva U3259916|contribs]]) 05:53, 10 August 2025 (UTC)

:{{ping|Eva U3259916}} For a thorough literature search, prioritise search of academic journal databases. Google Scholar provides very useful search. GenAI may be helpful for cross-checking and seeing whether it suggests any key literature that was missed during database searching. However, I would caution against relying on genAI as the primary tool for academic literature searching. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 06:39, 4 October 2025 (UTC)

== Proposal for rewriting this article  ==

The following is my proposed draft In its entirety: '''“Don’t.”''' [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 20:29, 29 April 2026 (UTC)
: Why? &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; 23:12, 20 June 2026 (UTC)
::I don’t need to explain why generative AI is bad. It’s already been done many, many times. [[Wikipedia|Our sister project]] is a good place to start. My “draft” is hyperbole, but it gets the point across. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 00:14, 21 June 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &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; 23:12, 20 June 2026 (UTC) and Alaexis</text>
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  <page>
    <title>User:Tommy Kronkvist</title>
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        <username>Tommy Kronkvist</username>
        <id>31941</id>
      </contributor>
      <comment>User statistics.</comment>
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      <text bytes="2646" sha1="4glxlwzvmf73xsbinricu9500dnjmh2" xml:space="preserve">&lt;div style=&quot;margin: 0 0 1em 0;&quot;&gt;{{userpage}}&lt;/div&gt;
{{Userboxtop|toptext=Babel:}}
{{#babel:sv|en-4|de-2|la-1}}
{{Userboxbottom}}
[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. &lt;small&gt;''Torminalis glaberrima'' (Gand.) Sennikov &amp; Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (June 21, 2026), I've made just over 394,100 edits to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]].

Swedish is my mother tongue&amp;nbsp;– even though I was born in Finland&amp;nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden.

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
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{{title|[[Motivation and emotion/Book/2026|&lt;big&gt;&lt;big&gt;Motivation and emotion&lt;/big&gt;&lt;/big&gt;]]:}}
&lt;!-- Sub-title and year --&gt;
&lt;div style=&quot;color: purple; font-size: large; font-weight: bold;&quot;&gt;
Understanding and improving our motivational and emotional lives using psychological science (2026)
&lt;/div&gt;
&lt;!-- Initial development --&gt;
{{notice|&lt;!-- Aiming for approximately ~150 topics - more coming.&lt;br&gt; --&gt;These topics will be&lt;!-- are being developed--&gt; developed by ~150 [[emerging scholar]]s as part of [[Motivation and emotion]], starting in August, 2026.}}
&lt;!-- Training resources --&gt;
{| style=&quot;border:2px solid #616F7C;background-color:WhiteSmoke;padding:2px;width:80%;margin: 0 auto 1em auto;&quot;
|-
|{{center top}}
;Training resources
[[Motivation and emotion/Assessment/Topic|Topic development]] |
[[Motivation and emotion/Assessment/Chapter|Book chapter]]

[[Motivation and emotion/Lectures/Introduction|Lecture 01]] | [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] |  [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] | [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]
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|-
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{| style=&quot;border:2px solid #616F7C;background-color:WhiteSmoke;padding:2px;width:80%;margin: 0 auto 1em auto;&quot;
|-
|{{center top}}
;How to select a topic
{{center bottom}}
&lt;!-- Pre-approved topics are listed below.&lt;br&gt; --&gt;
# [[Motivation and emotion/Wikiversity/Creating an account|Create an account and login]]
# Look through available topics without a current author
# Click &quot;Edit&quot; or &quot;Edit source&quot;
# Replace &quot;User Name&quot; for the topic of choice with your Wikiversity user name
# Publish the page
# Check that your user name appears correctly; if not, fix or [[Motivation and emotion/Help|get in touch]]
# Alternatively, [[Motivation and emotion/Assessment/Selection#New topics|negotiate a new topic]]
|-
|}
&lt;!-- Drafting message --&gt;
&lt;!-- These pages are undergoing a massive transformation.&lt;br&gt;~150 [[emerging scholar]]s who are studying [[Motivation and emotion|motivation and emotion]] are each authoring a resource about how psychological science can be used to understand and improve our lives.&lt;br&gt;Feel free to comment or contribute. --&gt;
&lt;!-- Marking message --&gt;
&lt;!--Most of the ~150 chapters have been submitted and are now undergoing expert review.&lt;br&gt;Feel free to continue improving and commenting.--&gt;

{{RoundBoxBottom}}&lt;!-- {{Motivation and emotion/Book/Quality3}} --&gt;&lt;noinclude&gt;[[Category:Motivation and emotion|{{SUBPAGENAME}}]]&lt;/noinclude&gt;</text>
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  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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      <text bytes="81244" sha1="l79lnnavrnc8xh541lbi3vrqgxbjw6u" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== The 5-cell 4-simplex ==

...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  
{{Clear}}

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

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

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.  
...

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>l79lnnavrnc8xh541lbi3vrqgxbjw6u</sha1>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="81391" sha1="4lpwqet50nxjzxrtqro3jzixjt0nsfu" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  
{{Clear}}


Notice that the 600-cell has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <revision>
      <id>2816343</id>
      <parentid>2816342</parentid>
      <timestamp>2026-06-20T14:59:26Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="81380" sha1="5l9mwpdujessyfgd0ivlb5y1sawfqia" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

Notice that the 600-cell has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="81390" sha1="magfbyfur6txnt4s4l71vouuxdmqlgg" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

Notice that the 600-cell has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

{{Clear}}
== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="81400" sha1="kyjnwenjb7tmws7agyxarxgs6psw34j" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

Notice that the 600-cell has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

{{Clear}}
== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="81399" sha1="e5rc20x0jyi1542wxn0orxgngcn9ab6" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

Notice that the 600-cell has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

{{Clear}}
== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

... 

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <revision>
      <id>2816347</id>
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      <timestamp>2026-06-20T15:09:03Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 5-cell 4-simplex */</comment>
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      <text bytes="81405" sha1="r7fvwczmvubk62vxi2ku1xrpuh29x8p" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

Notice that the 600-cell has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

{{Clear}}
== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

... 

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>r7fvwczmvubk62vxi2ku1xrpuh29x8p</sha1>
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      <text bytes="81390" sha1="2m88duyymb33pu2hp0hgmupxnekh82a" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

The table has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

{{Clear}}
== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

... 

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="81024" sha1="23znuu1c03pcdrm2y0n15ztl59nmsw5" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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;
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|&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 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

{{Clear}}
[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} &lt;small&gt;&lt;math&gt;r_{13}=\sqrt{1}&lt;/math&gt;&lt;/small&gt;]]
We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

The table has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

... 

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="80897" sha1="qstd15p9l273eemkmagft2ggwl27945" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

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

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

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

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

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

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

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

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

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

The table has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord does not occur in the 600-cell.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

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

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

... 

{{Clear}}

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <revision>
      <id>2816351</id>
      <parentid>2816350</parentid>
      <timestamp>2026-06-20T15:36:31Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 600-cell */</comment>
      <origin>2816351</origin>
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      <text bytes="80939" sha1="5a5t642kupgk195xt9xgxgq4ecp2n67" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&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 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

In the 600-cell this 90° isoclinic rotation in completely orthogonal great square planes takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

The table has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord, which is the edge of the regular 5-cell, does not occur in the 600-cell.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>5a5t642kupgk195xt9xgxgq4ecp2n67</sha1>
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    <revision>
      <id>2816353</id>
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        <username>Dc.samizdat</username>
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      <text bytes="80972" sha1="5bkqtud5wnw42fcrjqehjfktcgtxzux" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct 90° isoclinic rotation in completely orthogonal great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This rotation takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} polygon is a skew helix with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° {30/7} isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;14\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form five circular helixes of ten twisted parallel strands 5{ 24/10}=2{12/5} } that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

The table has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord, which is the edge of the regular 5-cell, does not occur in the 600-cell.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <revision>
      <id>2816368</id>
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      <text bytes="81054" sha1="3zyguifpzb4hgc4x7fyhe26h2sc7bco" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/8}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct 90° isoclinic rotation in great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This rotation takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each {30/7} edge chord makes seven 12° turns. The rotational curve over each {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form five circular helixes of ten twisted parallel strands 5{ 24/10}=2{12/5} } that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

The table has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord, which is the edge of the regular 5-cell, does not occur in the 600-cell.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="81064" sha1="oda9s5zqj2htjykto5ylaqzfydfe44k" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_3=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The &lt;math&gt;r_3=\sqrt{2}&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_3&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_5=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over &lt;math&gt;r_{5}=\sqrt{3}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. 

In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;r_8=\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over &lt;math&gt;r_8=\sqrt{2}&lt;/math&gt; isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_8&lt;/math&gt; chord is the 16-cell &lt;math&gt;r_3&lt;/math&gt; chord. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_8&lt;/math&gt; chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct 90° isoclinic rotation in great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This rotation takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each edge belonging to a distinct great square. The vertices of the invariant great squares of this rotation each make seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each {30/7} edge chord makes seven 12° turns. The rotational curve over each {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;r_{10}=\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can also rotate the 600-cell isoclinically  in the characteristic rotation of the 24-cell, by 60° in great hexagon planes, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The &lt;math&gt;r_{5}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_2&lt;/math&gt; chord, and the &lt;math&gt;r_{10}&lt;/math&gt; chord is the 24-cell &lt;math&gt;r_5&lt;/math&gt; chord. The rotational curve over each &lt;math&gt;r_{10}&lt;/math&gt; chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_{10}&lt;/math&gt; chords form five circular helixes of ten twisted parallel strands 5{ 24/10}=2{12/5} } that intersects each 600-cell vertex once. 

In the 600-cell this 60° isoclinic rotation in great hexagon planes takes place over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords, where it has period 30 and visits every vertex of a 600-cell Petrie polygon. The {30/11} polygon is a skew helix with each &lt;math&gt;r_{11}&lt;/math&gt; chord the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a distinct great hexagon. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {30/13} polygram of &lt;math&gt;r_{13}&lt;/math&gt; chords. The rotational curve over each &lt;math&gt;r_{13}&lt;/math&gt; chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; over &lt;math&gt;r_{13}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

In the 600-cell this characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;5\pi&lt;/math&gt; over 15 &lt;math&gt;r_5&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

{{Clear}}
[[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} &lt;small&gt;&lt;math&gt;r_{12}=\sqrt{3.618\sim}&lt;/math&gt;&lt;/small&gt;]]
In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of &lt;math&gt;r_{12}&lt;/math&gt; chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once.  

The table has an isoclinic rotation over every {30/''n''} polygon except {30/8}. The {30/8} chord, which is the edge of the regular 5-cell, does not occur in the 600-cell.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>oda9s5zqj2htjykto5ylaqzfydfe44k</sha1>
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  <page>
    <title>History of Cannibalism in China</title>
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    <id>327859</id>
    <revision>
      <id>2816389</id>
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        <username>Somme4096</username>
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      <minor />
      <comment>/* Han Dynasty */</comment>
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      <text bytes="278104" sha1="od0m08j5b088mh94i28hz1lfynu8h14" xml:space="preserve">{{Under construction}}

Among all major civilizations worldwide, China has the most recorded instances of cannibalism.&lt;ref&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社, 1994, &quot;中国封建时代的有关（食人习俗的）文字记载是极为丰富的。可以说，中国封建时代的食人习俗证据远比其他时代或其他国家为多&quot;&lt;/ref&gt; This entry documented 388 cannibalism cases recorded in 530 instances from the ''Twenty-Five Histories'' ([[w:Twenty-Four Histories|Twenty-Four Histories]] and [[w:Draft History of Qing|Draft History of Qing]]), consistent with prior research &lt;ref name=鄭麒來統計&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社，1994年版，第153-154页。&lt;/ref&gt;. According to another study, the [[w:Complete Classics Collection of Ancient China|Complete Classics Collection of Ancient China]], a comprehensive Chinese encyclopedic work, recorded 653 cases of filial piety act involving cutting own flesh to cure parents' illness&lt;ref name=鄭麒來統計/&gt;.

Several factors are generally considered responsible for this prevalence. 

* China experienced more famines than any other major civilizations.&lt;ref&gt;邓拓，《中国救荒史》，1937年，“我国灾荒之多，世界罕有，就文献可考的记载来看，从公元前十八世纪，直到公元二十世纪的今日，将近四千年间，几于无年无灾，也几乎无年无荒。西欧学者甚至称我国为‘饥荒的国度’（The Land of Famine)。” &lt;/ref&gt; 
* China experienced the most frequent and intense conflicts among major civilizations.&lt;ref&gt;秦晖，《中国历史上，何来如此深仇大恨》，“中国秦以后历代王朝的寿命不但比‘封建’时代的周‘王朝’和欧洲、日本的宗主王系（不是dynasty）短很多，其‘改朝换代’的巨大破坏性更几乎是人类历史上独有的。……世界史上别的民族有遭到外来者屠杀而种族灭绝的，有毁灭于庞贝式的自然灾变的，但像中国这样残忍的自相残杀确实难找他例。”&lt;/ref&gt; &lt;ref&gt; 福山《政治秩序的起源》，2014年，广西师范大学出版社，第7章，“与其他军事化社会相比，周朝的中国异常残暴。有个估计，秦国成功动员了其总人口的8%到20%，而古罗马共和国的仅1%，希腊提洛同盟的仅5.2%，欧洲早期现代则更低”&lt;/ref&gt; 
* Specific cultural beliefs developed in China, including:
** Rationalizing cannibalism as a means of expressing animosity&lt;ref&gt;《左传·襄公二十一年》，“然二子者，譬如禽兽，臣食其肉而寝处其皮矣”；岳飞，《满江红》，“壮志饥餐胡虏肉，笑谈渴饮匈奴血”；《三国演义》、《水浒传》多处有吃仇人肉的描写；等等&lt;/ref&gt;.
** Attributing medicinal properties to human flesh &lt;ref&gt;唐，陈藏器，《本草拾遗》；明，李时珍，《本草纲目》&lt;/ref&gt;.
** Viewing the practice of cutting own flesh to treat elder relatives as a noble demonstration of filial piety&lt;ref&gt; 《宋史· 卷四百五十六·列传第二百一十五·孝义》：“太祖、太宗以来，……刲股割肝，咸见褒赏；”&lt;/ref&gt;
* China established a comprehensive official historical record system early on, which remained functional even during periods of significant social chaos, preserving extensive historical documentation.

==Statistics==

Key-Ray Chong categorized records of cannibalism within the Twenty-Five Histories, based on their causes.&lt;ref name=&quot;鄭麒來統計&quot; /&gt;

{| class=&quot;wikitable&quot;
|-
!Historical Records!!Subtotal!!Wartime Famine!!Wartime Hatred!!Natural Disasters!!Peace-time Hatred!!Loyalty!!Filial Piety!!Taste!!Other
|-
| [[:w:Shiji|Records of the Grand Historian(''Shiji'')]]||19||6||11	|| ||2|| || || ||
|-
| [[:w:Book of Han|Book of Han]] ||25||11||1||13|| || || || ||
|-
| [[:w:Book of the Later Han|Book of the Later Han]]||26||15|| ||11 ||||||||||
|-
| [[:w:Records of the Three Kingdoms|Records of the Three Kingdoms]]||7||4|| ||3 ||||||||||
|-
| [[:w:Book of Jin|Book of Jin]]||32||16||1||13||2 ||||||||
|-
| [[:w:Book of Wei|Book of Wei]]||8||6||1||1	||||||||||
|-
| [[:w:History of the Southern Dynasties|History of the Southern Dynasties]]||18||12||3||3 ||||||||||
|-
| [[:w:History of the Northern Dynasties|History of the Northern Dynasties]]||6||3||3 ||||||||||||
|-
| [[:w:Book of Northern Qi|Book of Northern Qi]]||2||2 ||||||||||||||
|-
| [[:w:Book of Song|Book of Song]]||2||1||1 ||||||||||||
|-
| [[:w:Book of Liang|Book of Liang]]||9||5||2||2 ||||||||||
|-
| [[:w:Book of Chen|Book of Chen]]||1||1 ||||||||||||||
|-
| [[:w:Book of Sui|Book of Sui]]||8||2||3||3||||||||||
|-
| [[:w:Historical Records of the Five Dynasties|Historical Records of the Five Dynasties]]||15||10||4|| || || ||1||||
|-
| [[:w:Old History of the Five Dynasties|Old History of the Five Dynasties]]||5||3||1||1||||||||||
|-
| [[:w:History of Jin|History of Jin]]||3||||||3||||||||||
|-
| [[:w:History of Liao|History of Liao]]||1||||||1||||||||||
|-
| [[:w:History of Yuan|History of Yuan]]||46||5||1||27||||||13||||
|-
| [[:w:History of Song (book)|History of Song]]||43||4||4||14||||||20||1	||
|-
| [[:w:History of Ming|History of Ming]]||45||5||||22||||||17	||1||
|-
| [[w:Draft History of Qing|Draft History of Qing]]||76||3||||15	||||||58||||
|-
!Total!!397!!114!!36!!132!!4!!0!!109!!2	!!

|}

However, this statistics is incomplete and partially incorrect. It omitted [[:w:Book of Zhou|Book of Zhou]], [[:w:Book of Qi|Book of Southern Qi]], [[:w:Old Book of Tang|Old Book of Tang]], [[:w:New Book of Tang|New Book of Tang]] originally included in the ''Twenty-Five Histories,'' and failed to remove duplicated records in [[:w:History of Ming|History of Ming]].

In addition to previous research, Key-Ray Chong compiled 653 cases of filial piety act involving cutting one's own flesh to cure relatives in [[w:Complete Classics Collection of Ancient China|Complete Classics Collection of Ancient China]], of which 99% involved women, and 56% of these cases involved daughters-in-law cutting their own flesh for their mothers-in-law. Although this polarization may be the result of intentional selection bias, as both male and female cases of flesh-cutting to cure relatives are well documented in the ''Twenty-Five Histories.'' 

Key-Ray Chong concluded:&lt;ref&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社，1994年版，第5-8页。&lt;/ref&gt;

{{Blockquote|text=Chinese practice of survival cannibalism does not significantly differ from that of other cultures; However, &quot;learned cannibalism''(習得性食人)''&quot; in China earned unique characteristics, particularly in its historical prevalence and specific motivations.

Unlike many other regions, where religion played a central role in cannibalistic rituals, Chinese practices were largely secular, often driven by two emotional extremes: '''Virtue and Affection''', including acts performed out of loyalty (尽忠), filial piety (尽孝), or deep love. '''Vengeance and Hatred''', on the other hand, are acts performed for revenge (報仇), to wash away shames (雪恥), or out of pure animosity. To give an example, During wartimes, cannibalism was frequently practiced as a symbolic and literal act of consuming the enemy, rooted in deep-seated hatred.

It is worth noting that ''learned cannibalism'' was also associated with '''culinary appreciation''' or '''medicinal therapy''' among the upper classes. Human flesh was perceived as both a food source and a potent medicine, especially valued for enhancing sexual function. For example, Li Shizhen's [[:w:Compendium of Materia Medica|Compendium of Materia Medica]] listed 35 human organs or substances used for medicinal purposes.}}
==Xia, Zhou and Shang Dynasty==
Note that early Chinese history often blends myth with oral tradition. While these records lack contemporary archeological evidence, they are also historically significant as they reflect how later generations conceptualized the origins of social norms including cannibalism.

# c. 1940 BCE, Xia Dynasty
#: '''English:''' He [Houyi] relied on his archery and neglected civil affairs... The family retainers killed and boiled him, and fed him to his sons. His sons could not bear to eat him and died at city gate.
#: '''Original:''' {{lang|zh-cn|「……（后羿）恃其射也，不修民事而淫於原獸，棄武羅、伯因、熊髡、圉而用寒浞。……羿猶不悛，將歸自田，家眾殺而亨之。以食其子；其子不忍食諸，死於窮門。」}}
#: '''Source:''' ''Zuo Zhuan'', Chapter of Duke Xiang (《左傳·襄公》)
#  Reign of [[:w:King Weng of Zhou|King Weng of Zhou]], c.1112-1050 BCE 
#: '''English:'''　According to ''Diwang Shiji''(The Century of Emperors), [King] Zhou imprisoned King Wen(of Zhou Dynasty). King Wen's eldest son, Boyi Kao, was serving as a hostage in Yin and acted as a charioteer for King Zhou. King Zhou boiled [Boyi Kao] to make a meat soup and presented it to King Wen, saying: &quot;''A true sage should not eat a soup made of his own son.''&quot; 
#: King Wen ate it. King Zhou then remarked, &quot;Who was it said the Earl of the West (King Wen) was a sage? He ate a soup made of his own son without even realizing it.&quot;
#: '''Original:''' 「《帝王世紀》云，（紂）囚文王，文王之長子曰伯邑考，質於殷，為紂御。紂烹為羮，賜文王曰：聖人當不食其子羮。文王食之，紂曰，誰謂西伯聖者，食其子羮尚不知也。」
#: '''Source:''' Justice in History, book 3, records of Yin (《史記正義·卷三·殷本紀》)
#: '''Note:''' The ''Century of Emperors''(《帝王世紀》) cited above was written in [[:w:Jing Dynasty|Jin Period]], and the original is now lost.

== Spring and Autumn / Warring States Periods ==

The [[:w:Spring and Autumn period|Spring and Autumn]] and [[:w:Warring States period|Warring States]] periods (approx. 770–221 BC) marked a significant era where cannibalism was documented under various social and political motivations. Famous Chinese idioms such as &quot;exchanging children to eat&quot; (''易子而食'', from [[:w:Zuo Zhuan|Zuo Zhuan]]) and &quot;eating the flesh and sleeping on the skin&quot; (''食肉寝皮'', from [[:w:Zuo Zhuan|Zuo Zhuan]]) both originated during this time.

Cases of cannibalism during this period can be categorized into four dominant motivations.

# '''Warfare and Siege Famines:''' The most frequent cause. During prolonged sieges, resources were so depleted that citizens resorted to &quot;exchanging children to eat&quot; to avoid consuming their own offspring.
# '''Political motivation:''' A famous case is Yi Ya (易牙), who steamed his own son to serve as a delicacy to Duke Huan of Qi to prove his absolute loyalty.
# '''Intimidation:''' Cannibalism was used as a tool of terror or vengeance. Examples include the Di people killing and eating Duke Yi of Wei(''狄人殺食衛懿公''), or the Ruler of Zhongshan boiling the son of the his own general, Yue Yang(''中山君烹樂羊子''), to test his loyalty.
# '''Cultural customs:''' Early records mention peripheral groups, such as the &quot;People-Eating Kingdom&quot; (啖人國), though these may be the result of Han-centric view of &quot;barbaric&quot; outsiders.

While the [[:w:Zuo Zhuan|Zuo Zhuan]] records at least 15 major natural famines, there are no explicit records of cannibalism resulting from &quot;natural&quot; disasters during this specific period. 

However, historians often note that the absence of such records does not necessarily prove the absence of the practice; rather, it may reflect the selective focus bias on military and political events over lower-class sufferings.

=== Before Warring State period ===

# The practice of &quot;Yi Di&quot; (''宜弟'')
#: '''English''': In the ancient past, there was a kingdom called Kaishu to the east of Yue. When a first-born son was born, they would dismember and eat him. The practice is called &quot;Yi Di&quot; (meaning &quot;benefiting the younger brothers&quot;).
#: '''Original:''' 昔者越之東有輆沭之國者，其長子生，則解而食之，謂之「宜弟」。
#: '''Source:''' ''[[:w:Mozi|Mozi(Book)]]'', &quot;Moderation in Funerals&quot; (《墨子·節葬下》)
# Critique of &quot;Yi Di&quot;, by Mozi
#: '''English:''' Luyang Wenjun said to Mozi: &quot;South of Chu, there is a kingdom of man-eaters called Qiao. When a first-born son is born, they butcher and eat him, calling it 'Yi Di.' If the meat is flavorful, they present it to their ruler, who rewards the father. Is this not a detestable custom?&quot; 
#: Mozi replied: &quot;Even the customs of the Central Kingdoms are similar. How is killing a father and rewarding his son any different from eating a son and rewarding his father? If we do not govern by Benevolence and Righteousness, how can we criticize the barbarians for eating their sons?&quot;
#: '''Original:''' {{lang|zh-tw|魯陽文君語子墨子曰：「楚之南有啖人之國者橋，其國之長子生，則鮮而食之，謂之宜弟。美，則以遺其君，君喜則賞其父。豈不惡俗哉？」子墨子曰：「雖中國之俗，亦猶是也。殺其父而賞其子，何以異食其子而賞其父者哉？苟不用仁義，何以非夷人食其子也？」}}
#: '''Source:''' ''[[:w:Mozi|Mozi(Book)]]'', &quot;Lu Wen&quot; (《墨子·魯問》)
# Ethnographic Records of the Wuhu
#: '''English:''' To the west of the Nanman (Southern Barbarians) lies the Kingdom of Man-eaters, named [[:w:Cochin|Cochin]](Crossed rivers). There, man and woman bath in the same river, thus the name.
#: It is their custom to always dismember and eat the first-born son, calling it &quot;Yi Di.&quot; If the taste is delicious, they offer it to their ruler, who in turn rewards the father. Furthermore, if a man marries a beautiful wife, he offer her to his elder brother. These people are known today as the Wuhu.
#: '''Original:''' {{lang|zh-tw|其西有啖人國，生首子輒解而食之，謂之宜弟。味旨，則以遺其君，君喜而賞其父。取妻美，則讓其兄。今烏滸人是也。}}
#: '''Source:''' ''[[:w:Book of the Later Han|Book of the Later Han]]'', &quot;On the Southern and Southwestern Barbarians&quot; (《後漢書·南蠻西南夷列傳》)

=== In Warring State period ===
# During reigns of Duke Huan of Qi (''齊桓公'', r. 685–643 BCE)
#: '''English''': During the reign of Duke Huan of Qi, Yi Ya served the Duke as his personal chef. The Duke once said that he had never tasted steamed infant. Upon hearing this, Yi Ya steamed his own firstborn son and presented the dish to the Duke. Human nature is such that one loves one's own children; yet he who does not love his own son. Then, what he would do to his own lord?
#: '''Original:''' 夫易牙以调和事（齐桓）公，公曰&quot;惟蒸婴儿之未尝&quot;，于是蒸其首子而献之公。人情非不爱其子也，于子之不爱，将何有于公？
#: '''Source:''' ''[[:w:Guanzi (text)|Guanzi]]'', &quot;Minor Exaltation&quot; (《管子·小称》)
## Alternate records of &quot;Yi Ya&quot;, During reigns of Duke Huan of Qi (''齊桓公'', r. 685–643 BCE)
##: '''English''': Duke Huan of Qi was fond of rare delicacies, and so Yi Ya steamed his own son's head and presented it to him.
##: '''Original:''' 齐桓公好味，易牙蒸其子首而进之。
##: '''Source:''' ''[[:w:Han Feizi|Han Feizi]]'', &quot;The Two Handles&quot; (《韓非子·二柄·難一》)
# 660 BCE: The Death and Consumption of Duke Yi of Wei (''衛懿公'')
#: '''English''': The Di people arrived and overtook Duke Yi of Wei at Rongze, where they killed him. They consumed all of his flesh, only his liver was untouched.
#: '''Original:''' 狄人至，及（卫）懿公于荣泽，杀之，尽食其肉，独舍其肝。
#: '''Source:''' ''[[:w:Lüshi Chunqiu|Lüshi Chunqiu]]'' (《吕氏春秋》)
# 594 BCE: The Siege of Song
#: '''English''': The people of Song, fearing for their lives, sent Hua Yuan on a secret night mission into the Chu encampment. He climbed into the bed of Zi Fan and roused him, saying: &quot;Our lord has sent me, Yuan, to convey our dire situation: our city is reduced to trading children for food and splitting bones for fuel. Even so, a covenant made beneath the city walls — one that would mean the ruin of our state — we cannot accept. Withdraw thirty li (''unit of length, approx. 3 kilometers long)'' from us, and we will obey every command.&quot;
#: '''Original:''' 宋人惧，使华元夜入楚师，登子反之床，起之曰：&quot;寡君使元以病告，曰：'敝邑易子而食，析骸以爨。虽然，城下之盟，有以国毙，不能从也。去我三十里，唯命是听。'&quot;
#: '''Source:''' ''[[:w:Zuo Zhuan|Zuo Zhuan]]'', &quot;The Fifteenth Year of Duke Xuan&quot; (《左傳·宣公十五年》)
## 594 BCE: The Siege of Song  (alternate account)
##: '''English''': In the twentieth year of his reign, King Zhuang of Chu besieged Song in retaliation for the killing of a Chu envoy. After a siege of five months, the food supply within the city was completely exhausted. The inhabitants resorted to trading children for food and burning bones for fuel. Hua Yuan of Song went out to truthfully convey the situation to King Zhuang. The King said: &quot;Truly a man of virtue!&quot; and thereupon withdrew his forces.
##: '''Original:''' 二十年，（楚）围宋，以杀楚使也。围宋五月，城中食尽，易子而食，析骨而炊。宋华元出告以情。庄王曰：&quot;君子哉！&quot;遂罢兵去。
##: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Hereditary Houses of Chu, Vol. 40&quot; (《史記·卷四十·楚世家第十》)
# c. 500 BCE: Zhi the Robber (''盜跖'')
#: '''English''': Confucius and Liuxia Ji were friends; Liuxia Ji's younger brother was named Zhi the Robber. Zhi the Robber commanded a following of nine thousand men, swept through the empire with impunity, plundering the various lords. 
#: He stormed into dwellings, stole cattle and horses, and abducted women. Driven by greed, he cast aside all bonds of kinship, disregarding his parents and siblings, and made no offerings to his ancestors.
#: Wherever his forces passed, large states fortified their walls and small states withdrew into strongholds, and all the people suffered greatly. [...] At that time, Zhi the Robber was resting his men on the southern slope of Mount Tai, mincing human livers and eating them.
#: '''Original:''' 孔子与柳下季为友，柳下季之弟名曰盗跖。盗跖从卒九千人，横行天下，侵暴诸侯；穴室枢户，驱人牛马，取人妇女；贪得忘亲，不顾父母兄弟，不祭先祖。所过之邑，大国守城，小国入保，万民苦之。……盗跖乃方休卒徒太山之阳，脍人肝而餔之。
#: '''Source:''' ''[[:w:Zhuangzi (book)|Zhuangzi]]'', &quot;Robber Zhi&quot; (《莊子·盜跖》)
# 409 BCE: Yue Yang Drinks His Son's Broth
#: '''English''': Yue Yang served as a general of Wei and led an attack on Zhongshan. His son was residing in Zhongshan at the time, and the ruler of Zhongshan had the son boiled and sent the resulting broth to Yue Yang. Yue Yang sat beneath his campaign tent and drank it, finishing the entire cup. 
#: Marquis Wen of Wei said to his advisor Du Shize: &quot;Yue Yang, for my sake, ate the flesh of his own son.&quot; Du replied: &quot;One who can eat his own son's flesh. Who would he not eat?&quot; After Yue Yang had pacified Zhongshan, Marquis Wen rewarded his achievement but harbored doubts about his character.
#: '''Original:''' 乐羊为魏将而攻中山。其子在中山，中山之君烹其子而遗之羹，乐羊坐于幕下而啜之，尽一杯。文侯谓睹师赞曰：&quot;乐羊以我之故，食其子之肉。&quot;赞对曰：&quot;其子之肉尚食之，其谁不食？&quot;乐羊既罢中山，文侯赏其功而疑其心。
#: '''Source:''' ''[[:w:Zhanguo Ce|Zhanguo Ce]]'', &quot;Stratagems of Wei I, Vol. 22&quot; (《戰國策·卷二十二·魏策一》)
# 403 BCE: The Siege of Jinyang ''(晉陽之戰'')
#: '''English''': The three states of Zhi, Wei, and Han besieged Jinyang for over a year, and then diverted the Fen River to flood the city. The floodwaters rose to within three planks' breadth of the top of the walls. Within the city, cauldrons were suspended over fires for cooking, inhabitants exchanged children to eat.
#: '''Original:''' 三国（智魏韩）攻晋阳，岁馀，引汾水灌其城，城不浸者三版。城中悬釜而炊，易子而食。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Clan of Zhao, Vol. 43&quot; (《史記·卷四十三·趙世家第十三》)
## 403 BCE: The Siege of Jinyang (alternate record)
##: '''English''': The three clans of Zhi, Wei, and Han encircled the people of Zhao at Jinyang and flooded the city; the floodwaters rose to within three planks' breadth of the top of the walls, and the inhabitants resorted to eating men and horses.
##: '''Original:''' 三家（智魏韩）以国人围（赵国晋阳）而灌之，城不浸者三版，人马相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 1 (《資治通鑑·卷一》)
# 260 BCE: The Battle of Changping (''長平之戰'')
#: '''English''': By the ninth month, the Zhao soldiers had been without food for forty-six days, and in secret they began killing and ate each other.
#: '''Original:''' 至九月，赵卒不得食四十六日，皆内阴相杀食。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Bai Qi and Wang Jian, Vol. 73&quot; (《史記·卷七十三·白起王翦列傳第十三》)
## 260 BCE: The Battle of Changping (alternate record)
##: '''English''': The Zhao army was cut off from food for forty-six days, during which they secretly killed and ate each other.
##: '''Original:''' 赵军食绝四十六日，皆内阴相杀食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 5 (《資治通鑑·卷五》)
# 257 BCE: Li Tong(''李同'')'s Appeal at the Siege of Handan
#: '''English''': Li Tong said: &quot;The people of Handan are burning bones for fuel and trading children for food. Their plight could not be more desperate. Yet in your household, hundreds of concubines and maids are clothed in fine silk, with surplus grain and meat to spare, while the common people cannot complete a garment of coarse cloth and cannot fill themselves even with dregs and husks.&quot;
#: '''Original:''' 邯郸之民，炊骨易子而食，可谓急矣，而君之後宫以百数，婢妾被绮縠，馀粱肉，而民褐衣不完，糟糠不厌。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Lord Pingyuan and Yu Qing, Vol. 76&quot; (《史記·卷七十六·平原君虞卿列傳第十六》)
# c. 250 BCE: The Siege of Liaocheng
#: '''English''': Qi's general Tian Dan besieged Liaocheng for over a year, with heavy casualties among his troops, yet the city did not fall. Lu Zhonglian then composed a letter, tied it to an arrow, and shot it into the city, addressed to the Yan commander. The letter read: &quot;[...] Now you hold the exhausted people of Liaocheng against the full force of Qi's army — this is the defensive resolve of Mozi. Your men eat others and burn their bones for fuel, yet none harbor thoughts of surrender — this is the military discipline of Sun Bin. Your name shall be known throughout the realm.&quot;
#: '''Original:''' 齐田单攻聊城岁馀，士卒多死而聊城不下。鲁连乃为书，约之矢以射城中，遗燕将。书曰：……今公又以敝聊之民距全齐之兵，是墨翟之守也。食人炊骨，士无反外之心，是孙膑之兵也。能见於天下。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Lu Zhonglian and Zou Yang, Vol. 83&quot; (《史記·卷八十三·魯仲連鄒陽列傳第二十三》)

==Han Dynasty==
The wars between the Qin and Han dynasties caused large-scale famine and population decline across China, a pattern that would recur with nearly every subsequent dynastic transition.

# Early Han Dynasty: Famine and Cannibalism Following the Collapse of Qin
#: '''English''': At the founding of the Han dynasty, inheriting the devastation left by Qin, the various lords rose simultaneously in conflict. The people abandoned their livelihoods, and a great famine ensued. Price of one shi of rice reached five thousand coins; people ate each other, more than half the population perished. Emperor Gaozu then issued an order permitting the people to sell their children, and directed the starving to seek food in Shu and Han.
#: '''Original:''' 汉兴，接秦之敝，诸侯并起，民失作业而大饥馑。凡米石五千，人相食，死者过半。高祖乃令民得卖子，就食蜀、汉。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 205 BCE: Great Famine in Guanzhong, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': A great famine struck Guanzhong; the price of one hu of rice reached ten thousand coins, and people ate each other. The people were directed to seek food in Shu and Han.
#: '''Original:''' 关中大饥，米斛万钱，人相食。令民就食蜀、汉。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Gao, Vol. 1a&quot; (《漢書·卷一上·高帝紀第一上》)
## 205 BCE: Great Famine in Guanzhong, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': A great famine struck Guanzhong; the price of one hu of rice reached ten thousand coins, and people ate each other.
##: '''Original:''' 关中大饥，米斛万钱，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 9 (《資治通鑑·卷九》)
# 196 BCE: Minced flesh of Peng Yue, in ''[[:w:Records of the Grand Historian|Shiji]]''
#: '''English''': In the eleventh year, Empress Gao put to death the Marquis of Huaiyin; (Ying) Bu grew fearful at heart. In summer, Han executed Liang Wang Peng Yue, minced his flesh into paste, and sent portions of his flesh to all the lords. 
#:When it reached Huainan, the King of Huainan was out hunting; upon beholding the paste, he trembled greatly, and secretly ordered men to muster troops, watching for signs of trouble in the neighboring commanderies.
#: '''Original:''' 十一年，高后诛淮阴侯，布因心恐。夏，汉诛梁王彭越，醢之，盛其醢遍赐诸侯。至淮南，淮南王方猎，见醢，因大恐，阴令人部聚兵，候伺旁郡警急。
#: '''Source:''' ''[[:w:Records of the Grand Historian|Shiji]]'', &quot;Biography of Qing Bu&quot; (《史记·卷九十一·黥布列传第十三》)
# 138 BCE: Flood and Famine on the Yellow River Plain, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the third spring of that year, the Yellow River overflowed onto the Pingyuan plain. Great famine, people ate each other.
#: '''Original:''' 三年春，河水溢于平原，大饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Wu, Vol. 6&quot; (《漢書·卷六·武帝紀第六》)
## 138 BCE: Flood and Famine on the Yellow River Plain, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': The Yellow River overflowed onto the Pingyuan plain. Great famine, people ate each other.
##: '''Original:''' 河水溢于平原。大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 17 (《資治通鑑·卷十七》)
# 135 BCE: Ji An's Report on Famine in Henei, ''[[:w:Shiji|Shiji]]''
#: '''English''': Ji An returned and reported: &quot;A household fire has spread to neighboring houses. it is not worth undue concern. On my way, I passed through Henan, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another. I therefore took it upon myself, acting on temporary authority, to use the imperial tally to open the Henan granaries and relieve the destitute people. I now request to return the tally and submit to punishment for acting beyond my authority.&quot; The Emperor, recognizing his virtue, pardoned him.
#: '''Original:''' 还报曰：&quot;家人失火，屋比延烧，不足忧也。臣过河南，河南贫人伤水旱万余家，或父子相食，臣谨以便宜，持节发河南仓粟以振贫民。臣请归节，伏矫制之罪。&quot;上贤而释之。
#: '''Source:''' ''[[:w:Shiji|Shiji]]'', &quot;Biographies of Ji An and Zheng Dangshi, Vol. 120&quot; (《史記·卷一百二十·汲鄭列傳第六十》)
## 135 BCE: Ji An's Report on Famine in Henei, ''[[:w:Book of Han|Book of Han]]''
##: '''English''': [Ji An] returned and reported: &quot;A household fire has spread to neighboring houses — it is not worth undue concern. On my way, I passed through Henei, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another. I therefore took it upon myself, acting on temporary authority, to use the imperial tally to open the Henei granaries and relieve the destitute people. I request to return the tally and submit to punishment for acting beyond my authority.&quot; The Emperor, recognizing his virtue, pardoned him and transferred him to serve as Prefect of Xingyang.
##: '''Original:''' 还报曰：&quot;家人失火，屋比延烧，不足忧。臣过河内，河内贫人伤水旱万余家，或父子相食，臣谨以便宜，持节发河内仓粟以振贫民。请归节，伏矫制罚。&quot;上贤而释之，迁为荥阳令。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Zhang, Feng, Ji, and Zheng, Vol. 50&quot; (《漢書·卷五十·張馮汲鄭傳第二十》)
## 135 BCE: Ji An's Report, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': On my way, I passed through Henan, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another.
##: '''Original:''' 臣过河南，河南贫人伤水旱万馀家，或父子相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 17 (《資治通鑑·卷十七》)
# 114 BCE: Famine in Shandong, ''[[:w:Shiji|Shiji]]''
#: '''English''': At that time, the eastern provinces had suffered from Yellow River floods, and for several consecutive years the harvests had failed. In some places, spanning one to two thousand li, people resorted to eating one another.
#: '''Original:''' 是时山东被河灾，及岁不登数年，人或相食，方一二千里。
#: '''Source:''' ''[[:w:Shiji|Shiji]]'', &quot;Treatise on Equalization, Vol. 30&quot; (《史記·卷三十·平準書第八》)
## 114 BCE: Famine in Shandong(the East), ''[[:w:Book of Han|Book of Han]](1)''
##: '''English''': At that time, the eastern provinces had suffered from Yellow River floods, and for several consecutive years the harvests had failed. In some places, spanning two to three thousand li, people resorted to eating one another. The Emperor, moved by compassion, ordered the famine victims to travel and seek food in the Yangtze and Huai River regions, and those who wished to remain were permitted to settle there. Imperial envoys with carriages and canopies followed one another on the roads to escort them, and grain from Ba and Shu was dispatched to provide relief.
##: '''Original:''' 是时山东被河灾，乃岁不登数年，人或相食，方二三千里。天子怜之，令饥民得流就食江、淮间，欲留，留处。使者冠盖相属于道护之，下巴、蜀粟以赈焉。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24b&quot; (《漢書·卷二十四下·食貨志第四下》)
## 114 BCE: Famine in the East, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In the third month of the third Yuanding year, water froze; in the fourth month, snow fell. In more than ten commanderies east of the passes, people ate each other.
##: '''Original:''' 元鼎三年三月水冰，四月雨雪，关东十余郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on the Five Elements, Vol. 27&quot; (《漢書·卷二十七中之下·五行志第七中之下》)
## 114 BCE: Famine in the East, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': More than forty commanderies and kingdoms east of the passes suffered famine, people ate each other.
##: '''Original:''' 关东郡、国四十馀饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 20 (《資治通鑑·卷二十》)
# 113 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In summer, the fourth month, hail fell. In more than ten commanderies and kingdoms east of the passes, Great Famine; people ate each other.
#: '''Original:''' 夏四月，雨雹，关东郡国十余饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Wu, Vol. 6&quot; (《漢書·卷六·武帝紀第六》)
# 141–87 BCE: Critique of Emperor Wu's Reign, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': &quot;Though Emperor Wu had merit in driving back the four barbarians and expanding the realm, yet he slew great numbers of his men, exhausted the people's wealth, indulged in extravagance without measure. 
#: The realm was left hollow and depleted, the hundred folk scattered and adrift, half perished. Locusts rose in great swarms, scorching the earth for thousands of li; in some places people ate each other, and the stores have not recovered to this day. 
#: He bestowed no virtue nor grace upon the people, and ought not to have temple rites established in his honour.&quot;
#: '''Original:''' 武帝虽有攘四夷广土斥境之功，然多杀士众，竭民财力，奢泰亡度，天下虚耗，百姓流离，物故者半。蝗虫大起，赤地数千里，或人民相食，畜积至今未复。亡德泽于民，不宜为立庙乐。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Sui, Liang, Xiahou, Jing, Yi and Li, Vol. 75&quot; (《漢書·卷七十五·眭兩夏侯京翼李傳第四十五》)
# c. 104 BCE: Depletion of the Realm After Dong Zhongshu, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': After Zhongshu's death, expenditures grew ever greater, the realm was hollow and depleted, and once more people ate each other.
#: '''Original:''' 仲舒死后，功费愈甚，天下虚耗，人复相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In the ninth month, eleven commanderies and kingdoms east of the passes suffered great floods. Famine; in some places people ate each other. Neighboring commanderies were called upon to render aid in coin and grain.
#: '''Original:''' 九月，关东郡国十一大水，饥，或人相食，转旁郡钱、谷''(穀)''以相救。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Yuan, Vol. 9&quot; (《漢書·卷九·元帝紀第九》)
## 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In the first year of Chuyuan under Emperor Yuan, [...] in the fifth month the Bohai Sea overflowed greatly. In the sixth month, Great Famine struck the east; many among the people starved to death, and in Langye Commandery people ate each other.
##: '''Original:''' 元帝初元元年，……其五月，勃海水大溢。六月，关东大饥，民多饿死，琅邪郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Astronomy, Vol. 26&quot; (《漢書·卷二十六·天文志第六》)
## 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': In autumn, the ninth month, eleven commanderies and kingdoms east of the passes suffered great floods and famine; in some places people ate each other.
##: '''Original:''' 秋，九月，关东郡、国十一大水，饥，或人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 28 (《資治通鑑·卷二十八》)
# 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In the sixth month, famine struck the east; in the land of Qi, people ate each other.
#: '''Original:''' 六月，关东饥，齐地人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Yuan, Vol. 9&quot; (《漢書·卷九·元帝紀第九》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': When Emperor Yuan ascended the throne, great floods struck the realm; eleven eastern commanderies suffered most grievously. In the second year, famine struck the land of Qi; grain reached three hundred coins per shi, many among the people starved to death, and in Langye Commandery people ate each other.
##: '''Original:''' 元帝即位，天下大水，关东郡十一尤甚。二年，齐地饥，谷''(穀)''石三百余，民多饿死，琅邪郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](3)''
##: '''English''': The following year, in the second month, on the day wuwu, the earth shook. That summer, in the land of Liu, people ate each other. [...] Yi Feng memorialized: &quot;The eastern lands have suffered famine for years running, compounded by pestilence; the hundred folk are wan with hunger, and some have come to eat each other. The earth trembles repeatedly, the heavens are turbid, and the light of the sun grows dim.&quot;
##: '''Original:''' 明年二月戊午，地震。其夏，刘地人相食。……（翼奉）上疏曰：……今东方连年饥馑，加之以疾疫，百姓菜色，或至相食。地比震动，天气溷浊，日光侵夺。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Sui, Liang, Xiahou, Jing, Yi and Li, Vol. 75&quot; (《漢書·卷七十五·眭兩夏侯京翼李傳第四十五》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](4)''
##: '''English''': When Emperor Yuan first ascended the throne, he summoned Yu to serve as Remonstrant Counsellor and repeatedly sought his counsel on affairs of governance. At that time the harvests had failed and many commanderies were in distress. 
##: Yu exclaimed: &quot;Now the people die of Great Famine; the dead go unburied and are eaten by dogs and swine. People eat each other, whilst the horses in the imperial stables feed on grain and grow so fat and vigorous that they must be walked daily to work it off. Is this what it means for a sovereign, having received the Mandate of Heaven, to be father and mother to the people?&quot;
##: '''Original:''' 元帝初即位，征禹為諫大夫，數虛己問以政事。是時，年歲不登，郡國多困，禹奏言：[……] 今民大飢而死，死又不葬，為犬豬食。人至相食，而廄馬食粟，苦其大肥，氣甚怒至，乃日步作之。王者受命於天，為民父母，固當若此乎！（
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Wang, Gong, Liang Gong and Bao, Vol. 72&quot; (《漢書·卷七十二·王貢兩龔鮑傳第四十二》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](5)''
##: '''English''': Kuang Heng memorialized: &quot;The eastern lands have suffered famine for years running; the hundred folk are in want and distress, and some have come to eat each other. This hath all arisen from levies and taxes being too heavy, the burdens borne by the people being too great, and the officials failing in their duty to settle and succour them.&quot;
##: '''Original:''' 匡）衡上疏曰：……今关东连年饥馑，百姓乏困，或至相食，此皆生于赋敛多，民所共者大，而吏安集之不称之效也。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Kuang, Zhang, Kong and Ma, Vol. 81&quot; (《漢書·卷八十一·匡張孔馬傳第五十一》)
## 47 BCE: Famine in Qi, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Famine struck the east; in the land of Qi, people ate each other.
##: '''Original:''' 关东饥，齐地人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 28 (《資治通鑑·卷二十八》)
# 17 BCE: Emperor Cheng's Edict Dismissing Xue Xuan, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': Emperor Cheng decreed the dismissal of Xue Xuan, saying: &quot;I, being unenlightened, have seen repeated ill omens; the harvests have failed year upon year, the granaries stand empty, the hundred folk suffer Great Famine, wandering and scattered upon the roads. Those who have perished of pestilence number in the tens of thousands; people eat each other, bandits rise on all sides, and the offices of governance lie neglected. This is owing to mine own want of virtue and the failings of mine own ministers.&quot;
#: '''Original:''' 朕既不明，变异数见，岁比不登，仓廪空虚，百姓饥馑，流离道路，疾疫死者以万数，人至相食，盗贼并兴，群职旷废，是朕之不德而股肱不良也。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Xue Xuan and Zhu Bo, Vol. 83&quot; (《漢書·卷八十三·薛宣朱博傳第五十三》)
# 15 BCE: Floods in Liang and Pingyuan, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the second year of Yongshi, the kingdoms of Liang and Pingyuan suffered floods in consecutive years; people ate each other. The regional inspectors, prefects and chancellors were held accountable and dismissed.
#: '''Original:''' 永始二年，梁国、平原郡比年伤水灾，人相食，刺史、守、相坐免。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 14 CE: Great Famine Along the Frontier, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the first year of Tianfeng under Wang Mang, Great Famine struck the borderlands; people ate each other.
#: '''Original:''' 缘边大饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99b&quot; (《漢書·卷九十九中·王莽傳第六十九中》)
## 14 CE: Great Famine Along the Frontier, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Great Famine struck the borderlands; people ate each other.
##: '''Original:''' 缘边大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 37 (《資治通鑑·卷三十七》)
# 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In his final years, bandits rose in great numbers; armies were dispatched to suppress them, and their officers ran amok beyond the passes. In the northern borderlands and in the lands of Qing and Xu, people ate each other; east of Luoyang, grain reached two thousand coins per shi.
#: '''Original:''' 末年，盗贼群起，发军击之，将吏放纵于外。北边及青、徐地人相食，雒阳以东米石二千。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': [...] battle and slaughter, captivity by the four border peoples, criminal penalties, Great Famine, pestilence, and people eating each other had together reduced the households of the realm by half.
##: '''Original:''' 战斗死亡，缘边四夷所系虏，陷罪，饥疫，人相食，及莽未诛，而天下户口减半矣。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24b&quot; (《漢書·卷二十四下·食貨志第四下》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In that month, the Red Eyebrows slew the Grand Preceptor Xi Zhong Jing Shang. East of the passes, people ate each other.
##: '''Original:''' 是月，赤眉杀太师牺仲景尚。关东人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳第六十九下》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': East of the passes, people ate each other.
##: '''Original:''' 关东人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 38 (《資治通鑑·卷三十八》)
# 23 CE: The Fate of Wang Mang's Corpse, ''Book of Han''
#: '''English''': Wang Mang's severed head was sent to Gengshi and hung in the market of Wan. The common folk vied to strike and beat it; some cut out his tongue and ate it.
#: '''Original:''' 传（王）莽首诣更始，县宛市，百姓共提击之，或切食其舌。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳》)
# 23 CE: Siege of Wan — Cen Peng's Surrender, ''Hou Han Shu''
#: '''English''': When Zhen Fu fell and Cen Peng was wounded, he fled back to Wan and held the city together with Yan Shuo. Han forces besieged them for several months; the city's provisions were exhausted and people ate each other. Peng and Shuo thereupon surrendered the city.
#: '''Original:''' 汉兵攻之数月，城中粮尽，人相食，彭乃与说举城降。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Feng, Cen and Jia, Vol. 17&quot; (《後漢書·卷十七·馮岑賈列傳第七》)
# 23 CE: Siege of Wan — Cen Peng's Surrender, ''Zizhi Tongjian''
#: '''English''': [...] Han forces besieged them for several months. People within the city ate each other; they thereupon surrendered.
#: '''Original:''' 汉兵攻之数月，城中人相食，乃举城降。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 39 (《資治通鑑·卷三十九》)
# 24 CE: Li Xiong's Counsel to Gongsun Shu, ''Hou Han Shu''
#: '''English''': [...] &quot;Now the lands east of the mountains suffer Great Famine; the common folk eat each other. Where armies have passed, cities and towns are left as mounds of rubble.&quot;
#: '''Original:''' 今山东饥馑，人庶相食；兵所屠灭，城邑丘墟。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Wei Xiao and Gongsun Shu, Vol. 13&quot; (《後漢書·卷十三·隗囂公孫述列傳第三》)
# 25 CE: The Red Eyebrows Sack Chang'an, ''Book of Han''
#: '''English''': The Red Eyebrows burned the palaces and markets of Chang'an and slew Gengshi. The starving people ate each other; those who perished numbered in the hundreds of thousands. Chang'an was left a wasteland, and none walked its streets.
#: '''Original:''' 赤眉遂烧长安宫室市里，害更始。民饥饿相食，死者数十万，长安为虚，城中无人行。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳第六十九下》)
# 26 CE: Famine in Guanzhong, ''Hou Han Shu(1)''
#: '''English''': Great Famine struck Guanzhong; people ate each other.
#: '''Original:''' 关中饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Guangwu, Vol. 1a&quot; (《後漢書·卷一上·光武帝紀第一上》)
## 26 CE: Famine in Guanzhong, ''Hou Han Shu(2)''
##: '''English''': At that time, the three adjuncts were in great turmoil; people ate each other, the cities and towns were emptied, white bones lay strewn across the fields, and the survivors gathered here and there in fortified encampments, each holding firm.
##: '''Original:''' 时三辅大乱，人相食，城郭皆空，白骨蔽野，遗人往往聚为营保，各坚守不下。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Xuan and Liu Penzi, Vol. 11&quot; (《後漢書·卷十一·劉玄劉盆子列傳第一》)
## 26 CE: Famine in Guanzhong, ''Zizhi Tongjian''
##: '''English''': Great Famine struck the three adjuncts; people ate each other, the cities and towns were emptied, and white bones lay strewn across the fields.
##: '''Original:''' 三辅大饥，人相食，城郭皆空，白骨蔽野。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 40 (《資治通鑑·卷四十》)
# 27 CE: Siege of Ji, Zizhi Tongjian
#: '''English''': Within Zhu Fu's city of Ji, provisions were exhausted; people ate each other.
#: '''Original:''' 浮城中粮尽，人相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 41 (《資治通鑑·卷四十一》)
## 27 CE: Siege of Ji'', Hou Han Shu''
##: '''English''': Within Fu's city, provisions were exhausted; people ate each other.
##: '''Original:''' 浮城中粮尽，人相食。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Zhu, Feng, Yu, Zheng and Zhou, Vol. 33&quot; (《後漢書·卷三十三·朱馮虞鄭周列傳第二十三》)
# 27 CE: Yan Cen's Retreat to Nanyang, ''Hou Han Shu''
#: '''English''': At that time the people suffered Great Famine and ate each other; one jin of gold could be exchanged for but five sheng of beans. The roads were cut off and supplies could not get through; the soldiers subsisted on wild fruit.
#: '''Original:''' 时，百姓饥饿，人相食，黄金一斤易豆五升。道路断隔，委输不至，军士委以果实为粮。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Feng, Cen and Jia, Vol. 17&quot; (《後漢書·卷十七·馮岑賈列傳第七》)
# 109 CE: Great Famine in the Capital, ''Hou Han Shu''
#: '''English''': In the third month, Great Famine struck the capital; people ate each other.
#: '''Original:''' 三月，京师大饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor An, Vol. 5&quot; (《後漢書·卷五·孝安帝紀第五》)
## 109 CE: Great Famine in the Capital, ''Zizhi Tongjian''
##: '''English''': In the third month, Great Famine struck the capital; people ate each other.
##: '''Original:''' 三月，京师大饥，民相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 49 (《資治通鑑·卷四十九》)
# 109 CE: Floods and Famine Across the Realm, ''Hou Han Shu(1)''
#: '''English''': That year, the capital and forty-one commanderies and kingdoms suffered hail. Great Famine struck Bing and Liang; people ate each other.
#: '''Original:''' 是岁，京师及郡国四十一雨水雹。并、凉二州大饥，人相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor An, Vol. 5&quot; (《後漢書·卷五·孝安帝紀第五》)
## 109 CE: Floods and Famine Across the Realm, ''jin Shu''
##: '''English''': In the third year of Yongchu under Emperor An, floods and drought struck the realm; people ate each other.
##: '''Original:''' 安帝永初三年，天下水旱，人民相食。
##: '''Source:''' ''[[:w:Jin Shu|Jin Shu]]'', &quot;Treatise on Food and Commerce, Vol. 26&quot; (《晉書·卷二十六·志第十六·食貨》)
## 109 CE: Floods and Famine Across the Realm, ''Zizhi Tongjian''
##: '''English''': The capital and forty-one commanderies suffered floods; Great Famine struck Bing and Liang; people ate each other.
##: '''Original:''' 京师及郡国四十一雨水，并、凉二州大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 49 (《資治通鑑·卷四十九》)
# 151 CE: Drought and Famine, ''Hou Han Shu''
#: '''English''': Drought struck the capital. Great Famine afflicted Rencheng and Liang; people ate each other.
#: '''Original:''' 京师旱。任城、梁国饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Huan, Vol. 7&quot; (《後漢書·卷七·孝桓帝紀第七》)
## 151 CE: Drought and Famine, ''Zizhi Tongjian''
##: '''English''': Drought struck the capital; Great Famine afflicted Rencheng and Liang; people ate each other.
##: '''Original:''' 京师旱，任城、梁国饥，民相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 53 (《資治通鑑·卷五十三》)
# 155 CE: Famine in Sili and Jizhou, ''Hou Han Shu''
#: '''English''': In the second month, famine struck Sili and Jizhou; people ate each other.
#: '''Original:''' 二月，司隶、冀州饥，人相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Huan, Vol. 7&quot; (《後漢書·卷七·孝桓帝紀第七》)
# 155 CE: Famine in Sili and Jizhou, ''Zizhi Tongjian''
#: '''English''': In the second month, famine struck Sili and Jizhou; people ate each other.
#: '''Original:''' 二月，司隶、冀州饥，人相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 53 (《資治通鑑·卷五十三》)
# 170 CE: Spousal Cannibalism in Henei and Henan, ''Hou Han Shu''
#: '''English''': In the first month of spring in the third year of Jianning, in Henei wives ate their husbands, and in Henan husbands ate their wives.
#: '''Original:''' 三年春正月，河内人妇食夫，河南人夫食妇。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Ling, Vol. 8&quot; (《後漢書·卷八·孝靈帝紀第八》)
# 194 CE: Great Drought in the Three Adjuncts, ''Hou Han Shu''
#: '''English''': A great drought struck the three adjuncts from the fourth month to this day. At that time one hu of grain fetched fifty thousand coins, and one hu of beans or wheat twenty thousand. People ate each other; white bones lay heaped in piles.
#: '''Original:''' 三辅大旱，自四月至于是月。是时谷一斛五十万，豆麦一斛二十万，人相食啖，白骨委积。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Xian, Vol. 9&quot; (《後漢書·卷九·孝獻帝紀第九》)
## 194 CE: Great Drought in the Three Adjuncts, ''Zizhi Tongjian''
##: '''English''': From the fourth month no rain fell. One hu of grain was worth fifty thousand coins; within Chang'an, people ate each other.
##: '''Original:''' 自四月不雨至于是月，谷一斛直钱五十万，长安中人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# Liu Ping Spared by Cannibals, ''Hou Han Shu''
#: '''English''': Liu Ping, styled Gongzi, was a man of Pengcheng in Chu. During the upheavals of the Gengshi era, he and his mother hid together in the wilderness. 
#: One morning he went out to forage for food and was seized by starving bandits who meant to boil and eat him. He knelt and said: &quot;This morning I went to gather herbs for my aged mother, who depends on me for her life. I beg ye to let me return, feed my mother, and then come back to die.&quot; Tears streamed down his face. 
#: The bandits, moved by his sincerity, took pity and released him. Liu Ping returned, fed his mother, and then told her: &quot;I made a pledge to the bandits; honour forbids me to deceive them.&quot; He went back to the bandits. They were all greatly astonished and said to one another: &quot;We have long heard of men of fierce integrity — now we behold one. Go, friend; we have not the heart to eat thee.&quot; And so he was spared.
#: '''Original:''' 刘平字公子，楚郡彭城人也。[…] 更始时，天下乱，[…] 与母俱匿野泽中。平朝出求食，逢饿贼，将亨（通“烹”）之，平叩头曰：“今旦为老母求菜，老母待旷为命，愿得先归，食母毕，还就死。”因涕泣。贼见其至诚，哀而遣之。平还，既食母讫，因白曰：“属与贼期，义不可欺。”遂还诣贼。众皆大惊，相谓曰：“常闻烈士，乃今见之。子去矣，吾不忍食子。”于是得全。（《后汉书·卷三十九·刘赵淳于江刘周赵列传第二十九》）
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Zhao Xiao Offers Himself to Cannibals, ''Hou Han Shu''
#: '''English''': [After the fall of Wang Mang] the realm fell into turmoil and people ate each other. [Zhao Xiao's] younger brother Li was seized by starving bandits. 
#: Upon hearing this, Zhao Xiao bound himself and went to the bandits, saying: &quot;Li hath long been starved and is thin and gaunt; I filleth ye hunger better than him&quot; The bandits were greatly astonished and released them both, saying: &quot;Go home for now, and bring back rice and dried provisions instead.&quot; 
#: Xiao sought provisions but could find none; he returned to the bandits and offered himself for the pot. The bandits, marvelling at him, did him no harm.
#: '''Original:''' （王莽之後）天下乱，人相食。孝弟礼为饿贼所得，孝闻之，即自缚诣贼，曰：&quot;礼久饿羸瘦，不如孝肥饱。&quot;贼大惊，并放之，谓曰：&quot;可且归，更持米糒来。&quot;孝求不能得，复往报贼，愿就亨。众异之，遂不害。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Wang Lin Guards His Parents' Tomb, ''Hou Han Shu''
#: '''English''': In Runan there was a man named Wang Lin, a junior official, who lost his parents when he was but ten years of age. 
#: When great turmoil broke out and the people fled, only Wang Lin and his brothers remained to guard the burial mound, their weeping unceasing. His younger brother Ji went out and was seized by the Red Eyebrows, who meant to eat him. Wang Lin bound himself and begged to die in his brother's stead. 
#: The bandits, moved to pity, released them both; and by this deed Wang Lin's name became renowned throughout his hometown.
#: '''Original:''' 汝南有王琳巨尉者，年十余岁丧父母。因遭大乱，百姓奔逃，惟琳兄弟独守冢庐，号泣不绝。弟季，出遇赤眉，将为所哺，琳自缚，请先季死，贼矜而放遣，由是显名乡邑。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Wei Tan Spares His Fellow Captives, ''Hou Han Shu''
#: '''English''': Wei Tan of Langye, styled Shaoxian, was likewise seized by starved bandits. Several dozen captives were bound and awaited their turn to be boiled. 
#: The bandits, seeing that Tan appeared honest and trustworthy, set him apart to tend the cooking fire, though they bound him again each evening. Among the bandits was one Yi Changgong, who took especial pity on Tan; he secretly loosened Tan's bonds and said: &quot;Ye are all destined to be eaten; flee hence at once.&quot; 
#: Tan replied: &quot;I have tended the fire for ye, there I always had some leavings for myself; the others have been fed only on grass and weeds; better to eat (''relatively well-fed'') me instead.&quot; Changgong, moved by his righteousness, persuaded the others to release all the captives, and all were spared.
#: '''Original:''' 琅邪魏谭少闲者，时亦为饥寇所获，等辈数十人皆束缚，以次当亨（通“烹”）。贼见谭似谨厚，独令主爨，暮辄执缚。贼有夷长公，特哀念谭，密解其缚，语曰：&quot;汝曹皆应就食，急从此去。&quot;对曰：&quot;谭为诸君爨，恒得遗余，余人皆菇草莱，不如食我。&quot;长公义之，相晓赦遣，并得俱免。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Er Meng and Che Cheng Offer Themselves for Each Other, ''Hou Han Shu''
#: '''English''': Er Meng Ziming of Qi and Che Cheng Ziwei of Liangjun, brothers, were seized together by the Red Eyebrows and were about to be eaten. Meng and Cheng knelt and each begged to die in the other's stead. The bandits, moved to pity, released them both.
#: '''Original:''' 齐国兒萌子明、梁郡车成子威二人，兄弟并见执于赤眉，将食之，萌、成叩头，乞以身代，贼亦哀而两释焉。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Chunyu Gong Offers Himself for His Brother, ''Hou Han Shu''
#: '''English''': Chunyu Gong, styled Mengsun, was a man of Chunyu in Beihai. […] At the end of Wang Mang's reign, when famine and war arose, his elder brother Chong was seized by bandits who meant to boil and eat him. Gong begged to take his brother's place; both were released.
#: '''Original:''' 淳于恭字孟孙，北海淳于人也。[…] 王莽末，岁饥兵起，恭兄崇将为盗所亨，恭请代，得俱免。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)

== Three Kingdoms period ==
According to population statics at the time, the population of the Three Kingdoms period was only one-seventh of that during the reign of Emperor Huan of the Eastern Han Dynasty.&lt;ref&gt;秦晖，《中国历史上，何来如此深仇大恨》&lt;/ref&gt; This was the largest population decrease in Chinese history, evidenced by Cao Cao's poem; &quot;Pale bones exposed in wild fields, no crowing of roosters heard throughout thousands of li&quot; (白骨露于野，千里无鸡鸣).
# 194 CE: Famine During the Puyang Campaign, ''Sanguozhi''
#: '''English''': That year, one hu of grain fetched over fifty thousand coins; people ate each other. Newly recruited troops were thereupon disbanded.
#: '''Original:''' 是岁谷一斛五十余万钱，人相食，乃罢吏兵新募者。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Annals of Emperor Wu, Vol. 1&quot; (《三國志·卷一·魏書一·武帝紀》)
## 194 CE: Famine During the Puyang Campaign, ''Sanguozhi(2)''
##: '''English''': Cao Cao led his forces back and gave battle to Lü Bu at Puyang; his army fared ill and the two sides held their positions for over a hundred days. That year brought drought, locusts and scarcity of grain; the people ate each other. Lü Bu withdrew eastward to encamp at Shanyang.
##: '''Original:''' 太祖引军还，与布战于濮阳，太祖军不利，相持百余日。是时岁旱、虫蝗、少谷，百姓相食，布东屯山阳。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Lü Bu, Vol. 7&quot; (《三國志·卷七·魏書七·呂布臧洪傳》)
## 194 CE: Famine During the Puyang Campaign, ''Sanguozhi(3)''
##: '''English''': Cao Cao and Lü Bu held their positions at Puyang; Sima Lang thereupon led his household back to Wen. That year brought Great Famine; people ate each other. Lang gathered and succoured his kinsmen, tutored his younger brothers, and did not abandon his studies in that age of decline.
##: '''Original:''' 时岁大饥，人相食，朗收恤宗族，教训诸弟，不为衰世解业。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Sima Lang, Vol. 15&quot; (《三國志·卷十五·魏書十五·劉司馬梁張溫賈傳》)
## 194 CE: Famine During the Puyang Campaign, ''Hou Han Shu''
##: '''English''': Cao Cao heard of this and led his forces to attack Lü Bu; they fought repeatedly and held their positions for over a hundred days. That year brought drought, locusts and scarcity of grain; the people ate each other. Lü Bu withdrew to encamp at Shanyang.
##: '''Original:''' 曹操闻而引军击布，累战，相持百余日。是时，旱、蝗，少谷，百姓相食，布移屯山阳。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yan, Yuan Shu and Lü Bu, Vol. 75&quot; (《後漢書·卷七十五·劉焉袁術呂布列傳第六十五》)
# 194 CE: Cheng Yu's Human Jerky, Pei Songzhi's Commentary
#: '''English''': In the beginning, Cao Cao's forces lacked provisions. 
#: Cheng Yu seized supplies from his home county to provide three days' rations, mixed in no small part with dried human flesh. By this reason, he lost the favour of the ''(heavenly)'' court, and therefore never attained the rank of the Excellencies.
#: '''Original:''' 初，太祖乏食；昱略其本县，供三日粮，颇杂以人脯。由是失朝望，故位不至公。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Weijin Shiyu'', &quot;Biography of Cheng Yu, Vol. 14&quot; (裴松之《三國志注·卷十四·魏書十四·程昱傳》引《魏晉世語》)
# 195 CE: Great Famine at Chengshi, ''Sanguozhi''
#: '''English''': Cao Cao's forces were stationed at Chengshi. Great Famine; people ate each other.
#: '''Original:''' 太祖军乘氏，大饥，人相食。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Xun Yu, Vol. 10&quot; (《三國志·卷十·魏書十·荀彧荀攸賈詡傳》)
# 195 CE: The Siege of Dongjun, ''Hou Han Shu''
#: '''English''': [...] At first they still dug for rats and boiled sinew and hide; afterwards there was nothing left to eat.
#: The chief clerk reported that there were three dou of rice in the inner kitchen and requested it be made into gruel. Zang Hong said: &quot;How could I alone enjoy this?&quot; He had it made into thin porridge and distributed among all the troops.
#: He also slew all his beloved concubine to feed his officers and men. The officers and men all wept; none could raise their eyes to look upon him. Seventy or eighty men and women died lying upon one another; not one deserted or betrayed him.
#: '''Original:''' （东郡）初尚掘鼠，煮筋角，后无所复食，主簿启内厨米三斗，请稍为饘粥，洪曰：&quot;何能独甘此邪？&quot;使为薄糜，遍班士众。又杀其爱妾，以食兵将。兵将咸流涕，无能仰视。男女七八十人相枕而死，莫有离叛。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biography of Zang Hong, Vol. 58&quot; (《後漢書·卷五十八·虞詡等列傳》)
# 195 CE: The Siege of Dongjun, ''Zizhi Tongjian''
#: '''English''': At first they still dug for rats and boiled sinew and hide; afterwards there was nothing left to eat. 
#: The chief clerk reported only three sheng of rice in the inner kitchen and requested it be made into gruel. Zang Hong sighed: &quot;How could I alone enjoy this!&quot; He had it made into thin porridge and distributed among all the troops; he also slew his beloved concubine to feed his officers and men. 
#: The officers and men all wept; none could raise their eyes to look upon him. Seven or eight thousand men and women died lying upon one another; not one deserted or betrayed him.
#: '''Original:''' 初尚掘鼠煮筋角，后无可复食者。主簿启内厨米三升，请稍以为饘粥，臧洪叹曰：&quot;何能独甘此邪！&quot;使作薄糜，遍班士众，又杀其爱妾以食将士。将士咸流涕，无能仰视者。男女七八千人，相枕而死，莫有离叛者。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Hou Han Shu''
#: '''English''': [...] One hu of grain fetched fifty thousand coins; beans and wheat twenty thousand. People ate each other; white bones lay heaped in piles, and the stench of rot filled the roads. [...] After Li Jue and Guo Si turned upon each other and the Son of Heaven departed eastward, Chang'an stood empty for over forty days. The strong scattered; the weak ate each other. Within two or three years, not a human trace remained in Guanzhong.
#: '''Original:''' 自（李）傕、（郭）汜相攻，天子东归后，是时，谷一斛五十万，豆、麦二十万，人相食啖，白骨委积，臭秽满路。……长安城空四十余日，强者四散，蠃者相食，二三年间，关中无复人迹。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biography of Dong Zhuo, Vol. 72&quot; (《後漢書·卷七十二·董卓列傳第六十二》)
## 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Sanguozhi''
##: '''English''': At that time the three adjuncts still held several hundred thousand households. Li Jue and his confederates unleashed their troops to plunder, attacking and pillaging cities and towns. The people suffered Great Famine; within two years they had eaten each other to the last.
##: '''Original:''' 时三辅民尚数十万户，傕等放兵劫略，攻剽城邑，人民饥困，二年间相啖食略尽。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of Dong, the Two Yuans and Liu, Vol. 6&quot; (《三國志·卷六·魏書六·董二袁劉傳》)
## 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Jin Shu''
##: '''English''': [...] One hu of grain fetched fifty thousand coins; beans and wheat twenty thousand. People ate each other; white bones lay heaped in great mounds, the rotting remains befouling the roads. [...] Chang'an stood entirely empty; all scattered to the four winds. Within two or three years, not a traveller remained in Guanzhong. [...] Since Dong Zhuo's rebellion, the people had been scattered and adrift; grain reached over fifty thousand coins per shi, and many ate each other.
##: '''Original:''' 是时谷一斛五十万，豆麦二十万，人相食啖，白骨盈积，残骸余肉，臭秽道路。……长安城中尽空，并皆四散，二三年间，关中无复行人。……汉自董卓之乱，百姓流离，谷石至五十余万，人多相食。
##: '''Source:''' ''[[:w:Jin Shu|Jin Shu]]'', &quot;Treatise on Food and Commerce, Vol. 26&quot; (《晉書·卷二十六·志第十六·食貨》)
##195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Zizhi Tongjian''
##: '''English''': When Dong Zhuo first died, the three adjuncts still held several hundred thousand households. Li Jue and his confederates unleashed their troops to plunder; compounded by Great Famine, within two years the people had eaten each other nearly to the last. 
##: [...] At that time Chang'an stood empty for over forty days; the strong scattered, the weak ate each other, and within two or three years not a human trace remained in Guanzhong.
##: '''Original:''' 董卓初死，三辅民尚数十万户，李傕等放兵劫略，加以饥馑，二年间，民相食略尽。……是时，长安城空四十馀日，强者四散，羸者相食，二三年间，关中无复人迹。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# 195–197 CE: Wang Zhong the Cannibal, Pei Songzhi's Commentary
#: '''English''': Wang Zhong was a man of Fufeng who in his youth served as a village headman. When the three adjuncts fell into turmoil, Zhong, starving and desperate, ate human flesh, and followed a band of men southward toward Wuguan. [...]
#: The Master of the Wuguan Office, knowing that Zhong had once eaten human flesh, took him along on an imperial outing and had entertainers fasten a skull from a grave to Zhong's saddle, to the great amusement of all.
#: '''Original:''' 王忠，扶风人。少为亭长。三辅乱，忠饥乏噉人，随辈南向武关。……五官将知忠尝噉人，因从驾出行，令俳取冢间骷髅系著忠马鞍，以为欢笑。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Weilüe'', &quot;Annals of Emperor Wu, Vol. 1&quot; (裴松之《三國志注·魏書·武帝紀》引《魏略》)
# 196 CE: Liu Bei's Army Starves at Haixi, Zizhi Tongjian
#: '''English''': Liu Bei gathered his remaining forces and moved east to Guangling, gave battle to Yuan Shu, and was again defeated; he encamped at Haixi. Beset by hunger and hardship, his officers and men ate each other.
#: '''Original:''' 备收馀兵东取广陵，与袁术战，又败，屯于海西。饥饿困踧，吏士相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 62 (《資治通鑑·卷六十二》)
# 196 CE: Liu Bei's Army Starves at Haixi, Pei Songzhi's Commentary
#: '''English''': Liu Bei's army being at Guangling, hunger and hardship upon them; officers and men, high and low, ate each other.
#: '''Original:''' 備軍在廣陵，飢餓困踧，吏士大小自相啖食。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Yingxiong Ji'', &quot;Biography of the Progenitor Ruler, Vol. 32&quot; (裴松之《三國志注·卷三十二·蜀書·先主傳》引《英雄記》)
# 196 CE: Famine Under Gongsun Zan's Rule, ''Hou Han Shu''
#: '''English''': [...] That year brought drought and locusts; grain grew dear and people ate each other. Gongsun Zan, relying on his own abilities, showed no concern for the people.
#: '''Original:''' 是时，旱、蝗，谷贵，民相食。瓒恃其才力，不恤百姓。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yu, Gongsun Zan and Tao Qian, Vol. 73&quot; (《後漢書·卷七十三·劉虞公孫瓚陶謙列傳第六十三》)
# 197 CE: Famine Along the Yangtze and Huai, ''Hou Han Shu(1)''
#: '''English''': That year brought famine; along the Yangtze and Huai rivers, people ate each other.
#: '''Original:''' 是岁饥，江淮间民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Xian, Vol. 9&quot; (《後漢書·卷九·孝獻帝紀第九》)
## 197 CE: Famine Along the Yangtze and Huai, ''Hou Han Shu(2)''
##: '''English''': Yuan Shu's forces were weakened, his great generals dead, and his followers estranged and in revolt. Compounded by drought and failed harvests, his officers and people froze and starved; along the Yangtze and Huai, people had eaten each other nearly to the last.
##: '''Original:''' 术兵弱，大将死，众情离叛，加天旱岁荒，士民冻馁，江、淮间相食殆尽。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yan, Yuan Shu and Lü Bu, Vol. 75&quot; (《後漢書·卷七十五·劉焉袁術呂布列傳第六十五》)
## 197 CE: Famine Along the Yangtze and Huai, ''Sanguozhi''
##: '''English''': Yuan Shu's extravagance grew ever more excessive; his rear palace of several hundred consorts all wore fine silks, with surplus of grain and meat, whilst his officers and men froze and starved. Along the Yangtze and Huai the land was emptied; people ate each other.
##: '''Original:''' 荒侈滋甚，后宫数百皆服绮縠，余粱肉，而士卒冻馁，江淮间空尽，人民相食。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of Dong, the Two Yuans and Liu, Vol. 6&quot; (《三國志·卷六·魏書六·董二袁劉傳》)
## 197 CE: Famine Along the Yangtze and Huai, ''Zizhi Tongjian''
##: '''English''': Since the Zhongping era, the realm had fallen into turmoil; the people abandoned farming, armies rose on all sides, and provisions were ever wanting. When hungry, the troops plundered; when fed, they abandoned their surplus. Those who collapsed and scattered, undone by no enemy but themselves, were beyond counting. Yuan Shao in Hebei had his men subsist on mulberries; Yuan Shu along the Yangtze and Huai drew sustenance from cattail and river snails. The people ate each other, and the commanderies were left desolate.
##: '''Original:''' 中平以来，天下乱离，民弃农业，诸军并起，率乏粮谷，无终岁之计，饥则寇略，饱则弃馀，瓦解流离，无敌自破者，不可胜数。袁绍在河北，军人仰食桑椹。袁术在江淮，取给蒲蠃，民多相食，州里萧条。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 62 (《資治通鑑·卷六十二》)
# 238 CE: Siege of Xiangping. ''Sanguozhi''
#: '''English''': Gongsun Yuan was in dire stuation. His provisions exhausted, people ate each other, and the dead were very many.
#: '''Original:''' 渊窘急。粮尽，人相食，死者甚多。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of the Two Gongsuns, Tao and Four Zhangs, Vol. 8&quot; (《三國志·卷八·魏書八·二公孫陶四張傳》)
## 238 CE: Siege of Xiangping, ''Zizhi Tongjian''
##: '''English''': Gongsun Yuan was in dire situation; provisions in Xiangping were exhausted, people ate each other, and the dead were very many.
##: '''Original:''' 公孙渊窘急，粮尽，人相食，死者甚多。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 74 (《資治通鑑·卷七十四》)

==West Jin==

# 304 CE: The Famine of Chang'an and the Sack of Luoyang, ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': Shen Ju raised arms against Chang'an, yet was routed by (Sima) Yong. Zhang Fang greatly plundered Luo, then withdrew unto Chang'an. Thereupon the armies fell into dire want, and men did eat one another.
#: '''Original:''' 沈举举兵攻长安，为（司马）颙所败。张方大掠洛中，还长安。于是军中大馁，人相食。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals of Emperor Hui&quot; (《晋书·卷四·帝纪第四·惠帝》)
# 304 CE: The Plunder of Luoyang, in ''[[w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': (Zhang Fang) did seize from Luo above ten thousand bondsmen and bondswomen, both of state and private households, and marched them westward. The army, lacking victuals, did slay men and mingle their flesh with that of oxen and horses for sustenance.
#: '''Original:''' （张方）掠洛中官私奴婢万馀人而西。军中乏食，杀人杂牛马肉食之。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 85 (《资治通鉴》卷85)
# 306 CE: The Tyranny of Pan Tao and Bi Miao, in ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': (Pan) Tao and (Bi) Miao and their like seized (Sima) Yue and force him beyond the passes, falsely establishing a mobile administration, compelling the removal of ministers, issuing decrees by their own will, loosing soldiers to plunder and ravage, consuming the flesh of the common people, with corpses choking the roads and bleached bones filling the wilderness. Thus did the provincial lords betrayed their obligation, the cities and towns fall desolate, and the folk of Huai and Yu were casted into utter misery.
#: '''Original:''' （潘）滔、（毕）邈等劫（司马）越出关，矫立行台，逼徙公卿，擅为诏令，纵兵寇抄，茹食居人，交尸塞路，暴骨盈野。遂令方镇失职，城邑萧条，淮豫之萌，陷离涂炭。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biography of Zhou Jun et al.&quot; (《晋书·卷六十一·列传第三十一·周浚等》)
# 311 CE, eign of [[:w:Emperor Huai of Jin|Emperor Huai of Jin]]: The Rout at Ningping and the Death of Sima Yue, in ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': In the fifth year of Yongjia (the third month), (Sima) Yue did perish at Xiang. In the fourth month, Shi Le gave pursuit unto Ningping in Ku County; General Qian Duan sallied forth to resist him and fell in battle, the army breaking asunder. Thereupon Shi Le encircled the host of several hundred thousand with cavalry and loosed arrows upon them; the slain were heaped as mountains. Of princes, nobles, officers, and commoners, above a hundred thousand perished. Wang Mi's brother Zhang did burn the remnant and devour them. 
#: The people laid blame upon (Sima) Yue, and Emperor Huai issued a decree degrading Yue to the rank of a county king.
#: '''Original:''' 永嘉五年（三月），（司马越）薨于项。……（四月，）石勒追及于苦县宁平城，将军钱端出兵距勒，战死，军溃。……于是数十万众，（石）勒以骑围而射之，相践如山。王公士庶死者十余万。王弥弟璋焚其余众，并食之。天下归罪于（司马）越。（晋怀）帝发诏贬越为县王。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin (晉書)]]'', &quot;Biography of King Liang of Runan et al.&quot; (《晋书·卷五十九·列传第二十九·汝南王亮等》)
# 311 CE, Reign of [[:w:Emperor Huai of Jin|Emperor Huai of Jin]]: The Famine in the Passes, in ''[[w:Book of Jin|Book of Jin]](1)''
#: '''English''': At that time, famine ravaged the lands within the passes; the common folk consumed ate each other. Pestilence spreaded upon them, and bandits roamed openly, beyond the power of (Sima) Mo to suppress.
#: '''Original:''' 時關中饑荒，百姓相啖；加以疾疫，盜賊公行，（司马）模力不能制。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin (晉書)]]'', &quot;Biographies of the Imperial Clan&quot; (《晋书·卷三十七·列传第七·宗室》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](2)''
##: '''English''': Grand General Xun Xi memorialized to relocate the capital to Cangyuan; the Emperor was minded to comply, yet the great ministers, fearing (Pan) Tao, dared not carry out the edict, and the palace eunuchs, coveting their riches, were loath to depart. Famine grew great; people ate each other, and eight or nine in ten officials fled.
##: '''Original:''' 大将军苟晞表迁都仓垣,帝将从之,诸大臣畏滔,不敢奉诏,且宫中及黄门恋资财,不欲出。至是饥甚,人相食,百官流亡者十八九。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals, Emperor Huai &amp; Emperor Min&quot; (《晋书·卷五·帝纪第五·孝怀帝 孝愍帝》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](3)''
##: '''English''': By the Yongjia era, calamity and disorder had worsened greatly. East of Yongzhou, multitudes suffered hunger; they sold one another into bondage, and the wandering multitudes were beyond count. Six provinces — You, Bing, Si, Ji, Qin, and Yong — were struck by great locusts, devouring all grass, trees, and the fur of cattle and horses. Great pestilence followed, joined by famine. People were slain by brigands; corpses filled the rivers, and white bones covered the fields. As Liu Yao's forces pressed close, the court deliberated removing the capital to Cangyuan. People ate each other; famine and plague came together, and eight or nine in ten officials had fled.
##: '''Original:''' 至于永嘉,丧乱弥甚。雍州以东,人多饥乏,更相鬻卖,奔迸流移,不可胜数。幽、并、司、冀、秦、雍六州大蝗,草木及牛马毛皆尽。又大疾疫,兼以饥馑。百姓又为寇贼所杀,流尸满河,白骨蔽野。刘曜之逼,朝廷议欲迁都仓垣。人多相食,饥疫总至,百官流亡者十八九。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Treatise on Food and Commerce&quot; (《晋书·卷二十六·志第十六·食货》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](3)''
##: '''English''': Emperor Huai being besieged by Liu Yao, the imperial armies suffered repeated defeat, the treasury was exhausted, and the hundred officials were greatly famished; smoke of cooking fires was seen in no house. The starving fed upon one another. In the west, where Emperor Min resided, hunger was exceeding great; a peck of grain cost two taels of gold, and more than half the people perished.
##: '''Original:''' 怀帝为刘曜所围,王师累败,府帑既竭,百官饥甚,比屋不见火烟,饥人自相啖食。愍皇西宅,馁馑弘多,斗米二金,死者太半。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Treatise on Food and Commerce&quot; (《晋书·卷二十六·志第十六·食货》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](4)''
##: '''English''': When Luoyang fell into chaos, with thieves running rampant, people ate each other out of hunger. (Zhi) Yu, being ever poor and frugal, perished at last of starvation.
##: '''Original:''' 及洛京荒乱,盗窃纵横,人饥相食。虞素清贫,遂以馁卒。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Huangfu Mi et al.&quot; (《晋书·卷五十一·列传第二十一·皇甫谧等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](5)''
##: '''English''': (Wang) Mi, together with (Liu) Yao, attacked Xiangcheng and pressed upon the capital. The capital suffered a Great Famine; people ate each other, the common folk fled, and the dukes and ministers escaped to Heyin.
##: '''Original:''' 王弥后与曜寇襄城,遂逼京师。时京邑大饥,人相食,百姓流亡,公卿奔河阴。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Wang Mi et al.&quot; (《晋书·卷一百·列传第七十·王弥等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](6)''
##: '''English''': Wang Mi and Liu Yao arrived and joined (Huyan) Yan in besieging Luoyang. Within the city, famine was dire; people ate each other, the hundred officials scattered, and none held firm. The Xuanyang Gate fell; Mi and Yan entered the Southern Palace, ascended the Taiji Front Hall, and loosed their soldiers in great plunder, seizing all palace women and treasures. Yao thereupon slew all the princes, nobles, and officers below, in which numbered more than thirty thousand in all, and thereupon raised a great mound of their skulls north of the Luo River.
##: '''Original:''' 王弥、刘曜至,复与晏会围洛阳。时城内饥甚,人皆相食,百官分散,莫有固志。宣阳门陷,弥、晏入于南宫,升太极前殿,纵兵大掠,悉收宫人、珍宝。曜于是害诸王公及百官已下三万余人,于洛水北筑为京观。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Ere long, Luoyang fell to famine and distress; people ate each other, and eight or nine in ten officials had fled.
##: '''Original:''' 既而洛阳饥困,人相食,百官流亡者什八九。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 87 (《资治通鉴》卷87)
# 311 CE, Reign of Emperor Huai of Jin (永嘉五年): Great Famine and Cannibalism After the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': When Luoyang fell, Grand Commandant Xun Fan fled to Yangcheng, and General of the Guard Hua Hui fled to Chenggao. A Great Famine prevailed; the bandit chief Hou Du and his ilk seized men for food, and many of Fan's and Hui's followers were thus devoured.
#: '''Original:''' 及洛阳不守，太尉荀藩奔阳城，卫将军华荟奔成皋。时大饥，贼帅侯都等每略人而食之，藩、荟部曲多为所啖。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Shao Xu et al.&quot; (《晋书·卷六十三·列传第三十三·邵续等》)
# 312 CE: Cannibalism Among Han Zhao Troops, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': The Han Zhao generals Zhao Gu and Wang Sang, fearing absorption by Shi Le, sought to lead their forces back to Pingyang. Provisions within the army ran short, and soldiers ate each other.
#: '''Original:''' 汉安北将军赵固、平北将军王桑恐为石勒所并，欲引兵归平阳。军中乏粮，士卒相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 88 (《资治通鉴》卷88)
# 312 CE: Cannibalism in Shi Le's Army at Gepei, in ''[[:w:Book of Jin|Book of Jin]]'' and ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': (Shi) Le, at Gepei, built dwellings, encouraged farming, and constructed boats, intending to attack Jiankang. Yet wherever he marched, the people had fortified their walls and cleared the fields; nothing could be plundered, and great famine fell upon the army, so that soldiers ate each other.
#: '''Original:''' 勒于葛陂缮室宇，课农造舟，将寇建邺。……勒所过路次，皆坚壁清野，采掠无所获，军中大饥，士众相食。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Shi Le I&quot; (《晋书·卷一百四·载记第四·石勒上》)
# 312 CE: Cannibalism in Shi Le's Army at Gepei, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': As Shi Le marched north from Gepei, all along his path the people had fortified and cleared the fields; nothing could be seized. Famine within the army grew dire, and soldiers ate each other.
#: '''Original:''' 石勒自葛陂北行，所过皆坚壁清野，虏掠无所获，军中饥甚，士卒相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 88 (《资治通鉴》卷88)
# 314 CE: Monstrous Birth and Cannibalism in Guangyi, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': The wife of Yang Chong of Guangyi bore a child with two heads; her brother stole and ate it, and died within three days.
#: '''Original:''' 光义人羊充妻产子二头，其兄窃而食之，三日而死。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)
# 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Book of Jin|Book of Jin]](1)''
#: '''English''': In the tenth month of winter, the capital Chang'an suffered dire famine; a peck of grain cost two taels of gold, people ate each other, and more than half perished.
#: '''Original:''' 冬十月，京师饥甚，米斗金二两，人相食，死者太半。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals, Emperor Huai &amp; Emperor Min&quot; (《晋书·卷五·帝纪第五·孝怀帝 孝愍帝》)
## 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Book of Jin|Book of Jin]](2)''
##: '''English''': When Liu Yao again besieged the capital, (Suo) Chen and Qu Yun held fast to the inner city of Chang'an. Within, famine was dire; people ate each other, and the dead, fugitives, and deserters were beyond restraint; only the thousand loyal troops from Liangzhou stood firm unto death.
##: '''Original:''' 后刘曜又率众围京城、綝与麹允固守长安小城。……城中饥窘，人相食，死亡逃奔不可制，唯凉州义众千人守死不移。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Xie Xi et al.&quot; (《晋书·卷六十·列传第三十·解系等》)
## 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': In the eighth month, the Han Zhao Grand Marshal (Liu) Yao pressed upon Chang'an. Yao stormed the outer city; Qu Yun and Suo Chen withdrew to defend the inner city. All communication within and without was severed; famine within grew dire. A peck of grain cost two taels of gold, people ate each other, and more than half had perished; deserters and fugitives could not be restrained. Only the thousand loyal troops from Liangzhou stood firm. In the imperial granary there remained but several dozen cakes of leaven; Qu Yun ground them into gruel to feed the Emperor, yet ere long even these were exhausted.
##: '''Original:''' 八月，汉大司马曜逼长安。……曜攻陷長安外城，麴允、索綝退保小城以自固。內外斷絕，城中饑甚。斗米值金二兩，人相食，死者大半，亡逃不可制。唯涼州義眾千人守死不移。太倉有麴數十餅，麴允屑之為粥以供帝，既而亦盡。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 89 (《资治通鉴》卷89)
# 316 CE: Great Famine and Cannibalism in Beidi, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': Famine in Beidi was dire; people ate each other. Qiang Qiou's army transported grain to supply Qu Chang, but was defeated by Liu Ya.
#: '''Original:''' 北地饥甚，人相食啖，羌酋大军须运粮以给麹昌，刘雅击败之。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', Vol. 102 &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)

==East Jin==
# 319 CE: Slicing and Eating of Du Zeng's Flesh, ''Book of Jin''
#: '''English''': Du Zeng's forces collapsed; his generals Ma Jun and Su Wen captured him and surrendered to Zhou Fang. Zhou Fang wished to bring him alive to Wuchang, but Zhu Gui's son Zhu Chang and Zhao You's son Zhao Yin both begged for Du Zeng to avenge their fathers' grievances. Du Zeng was thereupon beheaded; Chang and Yin sliced his flesh and ate it.
#: '''Original:''' 曾众溃，其将马俊、苏温等执曾诣访降。访欲生致武昌，而朱轨息昌、赵诱息胤皆乞曾以复冤，于是斩杜曾，而昌、胤脔其肉而啖之。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 100, &quot;Biographies, Vol. 70: Wang Mi et al.&quot; (《晋书·卷一百·列传第七十·王弥等》)
# c. 321 CE: Xu Kan Fed to His Own Kin After Execution, ''Book of Jin''
#: '''English''': Shi Jilong attacked and captured Xu Kan, sending him to Xiangguo. Shi Le had him bagged and hurled to his death from the hundred-foot tower, then ordered the wives and children of Bu Du and others to disembowel and eat him; three thousand of Xu Kan's surrendered troops were buried alive.
#: '''Original:''' 石季龙攻陷徐龛，送之襄国，勒囊盛于百尺楼自上扑杀之，令步都等妻子刳而食之，坑龛降卒三千。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 105, &quot;Chronicles, Vol. 5: Shi Le, Part II et al.&quot; (《晋书·卷一百五·载记第五·石勒下等》)
# c. 337 CE: Shi Sui Slays Palace Women and Nuns, ''Book of Jin(1)''
#: '''English''': After Shi Sui assumed full governance, he abandoned himself to wine and lust, acting with arrogant depravity. He would roam the fields with music playing as he entered, or venture by night into the homes of court officials to violate their wives and concubines. 
#: Of the palace women whom he had adorned and found comely, he would behead them, wash away the blood, place their heads upon platters, and pass them round for viewing. He also brought in comely Buddhist nuns, defiled them, then slew them; their flesh was boiled together with beef and mutton and eaten, and portions were also distributed to his attendants, who were interested in the flavor.
#: '''Original:''' 邃自总百揆之后，荒酒淫色，骄恣无道，或盘游于田，悬管而入，或夜出于宫臣家，淫其妻妾。妆饰宫人美淑者，斩首洗血，置于盘上，传共视之。又内诸比丘尼有姿色者，与其交亵而杀之，合牛羊肉煮而食之，亦赐左右，欲以识其味也。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 106, &quot;Chronicles, Vol. 6: Shi Jilong, Part I&quot; (《晋书·卷一百六·载记第六·石季龙上》)
## c. 337 CE: Shi Sui Slays and Cooks Palace Women and Nuns, ''Zizhi Tongjian''
##: '''English''': Shi Sui, Crown Prince of Later Zhao, was arrogant, lustful, and cruel; he delighted in adorning comely consorts, beheading them, washing away the blood, placing their heads upon platters, and passing them amongst his guests for viewing. He further cooked their flesh and shared it for eating.
##: '''Original:''' 邃骄淫残忍，好妆饰美姬，斩其首，洗血置盘上，与宾客传观之，又烹其肉共食之。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 95 (《资治通鉴》卷95)
# 351 CE: Great Famine in Si and Ji Provinces, ''Book of Jin(1)''
#: '''English''': Bandits and rebels arose like swarms; a Great Famine struck Si and Ji Provinces; people ate each other. 
#: From the final years of Shi Jilong, Ran Min had dispersed all the granaries and treasuries to cultivate personal loyalty. Warfare with the Qiang and Hu raged without cease, with battles every month. 
#: The transplanted households of Qing, Yong, You, and Jing Provinces, together with the Di, Qiang, Hu, and Man peoples, numbering several hundred myriads, returned to their native lands; their routes met in one point, where all of they slaughtered and plundered one another. With famine and pestilence, only two or three in ten reached their destinations. Throughout the realm there was great disorder, and none remained to till the fields.
#: '''Original:''' 贼盗蜂起，司、冀大饥，人相食。自季龙末年而闵尽散仓库以树私恩。与羌胡相攻，无月不战。青、雍、幽、荆州徙户及诸氐、羌、胡、蛮数百余万，各还本土，道路交错，互相杀掠，且饥疫死亡，其能达者十有二三。诸夏纷乱，无复农者。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 107, &quot;Chronicles, Vol. 7: Shi Jilong, Part II&quot; (《晋书·卷一百七·载记第七·石季龙下》)
## 351 CE: Great Famine in Si and Ji Provinces, ''Zizhi Tongjian''
##: '''English''': The several hundred myriad transplanted peoples of Qing, Yong, You, and Jing Provinces — along with the Di, Qiang, Hu, and Man — whom Later Zhao had relocated, found the laws of Zhao no longer enforced and each returned to their native lands.
##: Their routes met in one point, where all of they slaughtered and plundered one another; only two or three in ten reached their destinations. The Central Plains fell into great disorder. Famine and pestilence followed; people ate each other, and none remained to till the fields.
##: '''Original:''' 后赵所徙青、雍、幽、荆四州人民及氐、羌、胡蛮数百万口，以赵法禁不行，各还本土；道路交错，互相杀掠，其能达者什有二、三。中原大乱。因以饥疫，人相食，无复耕者。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 99 (《资治通鉴》卷99)
# 352 CE: Famine in Ye, ''Book of Jin''
#: '''English''': Famine struck Ye; people ate each other. The palace women from the time of Shi Jilong were nearly all consumed.
#: '''Original:''' 邺中饥，人相食，季龙时宫人被食略尽。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 107, &quot;Chronicles, Vol. 7: Shi Jilong, Part II&quot; (《晋书·卷一百七·载记第七·石季龙下》)
## 352 CE: Famine in Ye'', Zizhi Tongjian''
##: '''English''': A Great Famine struck Ye; people ate each other. The palace women from the time of the former Zhao were nearly all consumed.
##: '''Original:''' 邺中大饥，人相食，故赵时宫人被食略尽。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 99 (《资治通鉴》卷99)
# 356 CE: Siege of Duan Kan's City, ''Zizhi Tongjian''
#: '''English''': Duan Kan defended the Yin city under siege; the roads for gathering firewood were cut off, and people ate each other within the city.
#: '''Original:''' 段龛婴城自守，樵采路绝，城中人相食。
#: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 100 (《资治通鉴·卷一百》)
# 385 CE: Great Famine at Chang'an, ''Book of Jin''
#: '''English''': At this time there was a Great Famine in Chang'an; people ate each other, and the generals, upon returning home, spat out flesh to feed their wives and children.
#: '''Original:''' 时长安大饥，人相食，诸将归而吐肉以饴妻子。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
## 385 CE: Great Famine at Chang'an, ''Wei Shu''
##: '''English''': Great Famine in Chang'an; people ate each other. Yao Chang rebelled at Beidi and allied with [Murong] Chong, jointly attacking Chang'an.
##: '''Original:''' 长安大饥，人民相食。姚苌叛于北地，与冲连和，合攻长安。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 95 &quot;Biographies 83, Liu Cong of the Xiongnu et al.&quot; (《魏书·卷九十五·列传第八十三·匈奴刘聪等》)
## 385 CE: Great Famine at Chang'an, ''Zizhi Tongjian''
##: '''English''': In the first month, [Former] Qin's [Fu] Jian held a banquet for his ministers. Chang'an was then stricken by famine; people ate each other, and the generals, upon returning home, spat out flesh to feed their wives and children.
##: '''Original:''' 正月，(前)秦(苻)堅朝饗群臣，時長安飢，人相食，諸將歸，吐肉以飼妻子。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 106 (《资治通鉴·卷一百零六》)
# 385 CE: Murong Chong's Forces Eat the Slain, ''Book of Jin''
#: '''English''': [Murong] Chong further dispatched his Secretariat Director Gao Gai to lead troops in a night assault on Chang'an, breaching the southern gate and entering the southern city. General of the Left Dou Chong and General of the Front Guards Li Bian and others repelled them, beheading 1,800 men, and divided the corpses for consumption.
#: '''Original:''' （慕容）冲又遣其尚书令高盖率众夜袭长安，攻陷南门，入于南城。左将军窦冲、前禁将军李辩等击败之，斩首千八百级，分其尸而食之。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
# 385 CE: Famine in You and Ji Prefectures, ''Book of Jin''
#: '''English''': Murong Gui's troops suffered greatly from hunger and many fled to Zhongshan; the people of You and Ji prefectures ate each other. Earlier, a popular rhyme in the Pass East had said: &quot;Youzhou — born to be destroyed; if not destroyed, the people shall be extinguished.&quot; This was [Murong] Cui's birth name. Having held out against [Fu] Pi for a full year, the common people were nearly all dead.
#: '''Original:''' 慕容垂军人饥甚，多奔中山，幽、冀人相食。初，关东谣曰：&quot;幽州，生当灭。若不灭，百姓绝。&quot;（慕容）垂之本名。与（符）丕相持经年，百姓死几绝。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
## 385 CE: Famine in You and Ji Prefectures, ''Zizhi Tongjian''
##: '''English''': Yan and Qin having held out against each other for a full year, You and Ji prefectures suffered a Great Famine; people ate each other, and settlements lay desolate. Many of Yan's soldiers starved to death; the King of Yan, [Murong] Cui, forbade the people from raising silkworms and had them subsist on mulberry berries.
##: '''Original:''' 燕、秦相持經年，幽、冀大饑，人相食，邑落蕭條，燕之軍士多餓死，燕王（慕容）垂禁民養蠶，以桑椹為食。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 106 (《资治通鉴·卷一百零六》)
# 386 CE: Fu Deng's Army Eats the Slain, ''Book of Jin''
#: '''English''': [Fu] Deng, having succeeded Wei Ping, thenceforth held sole command of military campaigns. At this time drought brought widespread hunger, and the roads were lined with the starving dead. Whenever Deng won a battle and slew the enemy, he called it &quot;cooked meat,&quot; and said to his men: &quot;You fight in the morning and by evening are sated with flesh — why fear hunger!&quot; The troops followed his lead, eating the flesh of the slain, and were thereby well-fed and fit for battle.
#: '''Original:''' （苻）登既代卫平，遂专统征伐。是时岁旱众饥，道殣相望，登每战杀贼，名为熟食，谓军人曰：&quot;汝等朝战，暮便饱肉，何忧于饥！&quot;士众从之，啖死人肉，辄饱健能斗。 
#: '''Source:''' [[wikipedia:Book of Jin|''Book of Jin'']], Vol. 115 &quot;Chronicles 15, Fu Pi et al.&quot; (《晋书·卷一百十五·载记第十五·苻丕等》)
# 387 CE: Famine in Jiuquan, ''Book of Jin''
#: '''English''': Wang Mu seized Jiuquan by surprise and proclaimed himself General-in-Chief and Governor of Liangzhou. At this time grain prices soared; one dou fetched five hundred cash, people ate each other, and more than half perished.
#: '''Original:''' 王穆袭据酒泉，自称大将军、凉州牧。时谷价踊贵，斗直五百，人相食，死者太半。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 122 &quot;Chronicles 22, Lü Guang et al.&quot; (《晋书·卷一百二十二·载记第二十二·吕光等》)
# 387 CE: Famine in Liangzhou, ''Zizhi Tongjian''
#: '''English''': Great Famine in Liangzhou; one dou of rice fetched five hundred cash, people ate each other, and more than half perished.
#: '''Original:''' 涼州大饑，米斗直錢五百，人相食，死者太半。
#: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 112 (《资治通鉴·卷第一百一十二》)
# c. 399 CE: Sun En Rebellion, ''Song Shu''
#: '''English''': In this time all means of livelihood were exhausted and the weak and elderly were many; the eastern lands suffered famine, and people exchanged children to eat. 
#: '''Original:''' 时生业已尽，老弱甚多，东土饥荒，易子而食;
#: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 100 &quot;Biographies 60, Preface&quot; (《宋书·卷一百·列传第六十·自序》)
## c. 399 CE: Sun En Rebellion, ''Wei Shu''
##: '''English''': When [Sun] En raised his rebellion, all eight commanderies became a field of carnage. … The rebels' prohibitions went unheeded; they killed at will, and the number of officers and commoners slain was beyond reckoning. Some county magistrates were pickled and fed to their own wives and children; those who refused were dismembered. Such was their cruelty.
##: '''Original:''' （孙）恩既作乱，八郡尽为贼场，……贼等禁令不行，肆意杀戮，士庶死者不可胜计，或醢诸县令以食其妻子，不肯者辄支解之，其虐如此。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 96 &quot;Biographies 84, the Usurper Jin's Sima Rui et al.&quot; (《魏书·卷九十六·列传第八十四·僭晋司马叡等》)
# 401 CE, Longan 5: Omen of Famine and Usurpation, ''Book of Jin''
#: '''English''': Huan Xuan's memorial arrived, defying imperial intent and affronting the throne. Thereafter Xuan usurped the throne, threw the capital into disorder; there was a Great Famine, people ate each other, and the common people fled — all were fulfillments of these omens.
#: '''Original:''' 九月，桓玄表至，逆旨陵上。其后玄遂篡位，乱京都，大饥，人相食，百姓流亡，皆其应也。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 13 &quot;Treatises 3, Astronomy III&quot; (《晋书·卷十三·志第三·天文下》)
# 402 CE: Famine at Guzang, ''Book of Jin''
#: '''English''': Grain prices at Guzang soared; one dou fetched five thousand cash, people ate each other, and over a hundred thousand starved to death. The city gates were shut by day, and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the barbarians. [Lü] Long, fearing this would demoralize the populace, had them all buried alive, corpses piled up and filled the streets.
#: '''Original:''' 姑臧谷价踊贵，斗直钱五千文，人相食，饿死者十余万口。城门昼闭，樵采路绝，百姓请出城乞为夷虏奴婢者日有数百。隆惧沮动人情，尽坑之，于是积尸盈于衢路。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 122 &quot;Chronicles 22, Lü Guang et al.&quot; (《晋书·卷一百二十二·载记第二十二·吕光等》)
## 402 CE: Famine at Guzang, ''Wei Shu''
##: '''English''': Juqu Mengxun and Tufa Rutan attacked repeatedly, leaving the people of Hexi unable to farm to the west. Grain prices soared; one dou fetched five thousand cash, people ate each other, and over a thousand starved to death. The city gates of Guzang were shut by day and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the barbarians. [Lü] Long, fearing this would demoralize the populace, had them all buried alive.
##: '''Original:''' 沮渠蒙逊、秃发辱檀频来攻击，河西之民，不得农西，谷价涌贵，斗直钱五千文，人相食，饿死者千余口。姑臧城门昼闭，樵采路断，民请出城，乞为夷虏奴婢者，日有数百。隆恐沮动人情，尽坑之。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 95 &quot;Biographies 83, Liu Cong of the Xiongnu et al.&quot; (《魏书·卷九十五·列传第八十三·匈奴刘聪等》)
## 402 CE: Famine at Guzang, ''Zizhi Tongjian''
##: '''English''': Great Famine at Guzang; one dou of rice fetched five thousand cash, people ate each other, and over a hundred thousand starved to death. The city gates were shut by day, and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the Hu barbarians; Lü Long, loathing the effect on morale, had them all buried alive, corpses piled up and filled the roads.
##: '''Original:''' 姑臧大饥，米斗直钱五千，人相食，饥死者十馀万口。城门昼闭，樵采路绝，民请出城为胡虏奴婢者，日有数百，吕隆恶其沮动众心，尽坑之，积尸盈路。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 112 (《资治通鉴·卷一百一十二》)
# 402 CE: Astronomical Omen of Famine, ''Book of Jin''
#: '''English''': In the fourth month, on the day xinsi, the moon occluded Mercury. In the seventh month, Great Famine; people ate each other.
#: '''Original:''' 元兴元年四月辛丑，月奄辰星。七月，大饥，人相食。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 12 &quot;Treatises 2, Astronomy II&quot; (《晋书·卷十二·志第二·天文中》)
## 402 CE: Famine in the Eastern Regions, ''Book of Jin(1)''
##: '''English''': In the seventh month of Yuanxing 1, Great Famine; people ate each other. Six or seven in ten east of the Zhe River died or fled; the population of Wu Commandery and Wuxing was halved, and tens of thousands more fled westward.
##: '''Original:''' 元兴元年七月，大饥，人相食。浙江以东流亡十六七，吴郡、吴兴户口减半，又流奔而西者万计。
##: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 13 &quot;Treatises 3, Astronomy III&quot; (《晋书·卷十三·志第三·天文下》)
## 402 CE: Famine in the Eastern Regions, ''Song Shu''
##: '''English''': In the seventh month [of Yuanxing 1], Great Famine; people ate each other. Six or seven in ten east of the Zhe River starved to death or fled; the population of Wu Commandery and Wuxing was halved, and tens of thousands more fled westward.
##: '''Original:''' （元兴元年）七月，大饥，人相食。浙江东饿死流亡十六七，吴郡、吴兴户口减半；又流奔而西者万计。
##: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 25 &quot;Treatises 15, Astronomy III&quot; (《宋书·卷二十五·志第十五·天文三》)
# 402 CE Kong Clan Distributes Grain, ''Song Shu''
#: '''English''': After the Sun En rebellion, the eastern lands suffered famine; people ate each other. The Kong clan distributed their household grain to relieve the neighbourhood, saving many lives; those who bore children thereafter named them Kong in gratitude.
#: '''Original:''' 及孙恩乱后，东土饥荒，人相食，孔氏散家粮以赈邑里，得活者甚众，生子皆以孔为名焉。
#: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 81 &quot;Biographies 41, Liu Xiuzhi et al.&quot; (《宋书·卷八十一·列传第四十一·刘秀之等》)
## 402 CE: Kong Clan Distributes Grain, ''Nan Shi''
##: '''English''': After the Sun En rebellion, the eastern lands suffered famine; people ate each other. The Kong clan distributed their household grain to relieve the neighbourhood, saving many lives; those who bore children thereafter named them Kong in gratitude.
##: '''Original:''' 孙恩乱后，东土饥荒，人相食，孔氏散家粮以振邑里，得活者甚众，生子皆以孔为名焉。
##: '''Source:''' [[:w:Nan Shi|''Nan Shi'']], Vol. 35 &quot;Biographies 25, Liu Zhan et al.&quot; (《南史·卷三十五·列传第二十五·刘湛等》)
# 409 CE: Cannibalism as Punishment for Regicide, ''Bei Shi''
#: '''English''': [Tuoba] Shao, together with several attendants and eunuchs, scaled the palace walls and violated the forbidden precinct. The Emperor [Daowu of Northern Wei, Tuoba Gui] started up in alarm, reached for his bow and sword but could not find them, and died suddenly. … The guards seized and delivered Shao. Thereupon Shao and his mother were put to death; the attending eunuchs and palace women who had acted as inner accomplices, numbering over ten, were executed. Those who had first laid hands upon the imperial person were carved alive and eaten by the assembled ministers on the main avenue south of the city.
#: '''Original:''' （拓跋）绍乃与帐下及宦者数人逾宫犯禁。帝（北魏道武皇帝拓跋珪）惊起，求弓刀不及，暴崩。……卫士执送绍，于是赐绍母子死，诛帐下阉官、宫人为内应者十数人。其先犯乘舆者，群臣于城南都街生脔食之。
#: '''Source:''' [[:w:Bei Shi|''Bei Shi'']], Vol. 16 &quot;Biographies 4, The Seven Princes of Daowu et al.&quot; (《北史·卷十六·列传第四·道武七王等》)
## 409 CE: Cannibalism as Punishment for Regicide, ''Wei Shu''
##: '''English''': The Supreme Ancestor (Taizong) arrived at the west of the city; the guards seized and delivered Shao. Thereupon Shao and his mother were put to death; the attending eunuchs and palace women who had acted as inner accomplices, numbering over ten, were executed. Those who had first laid hands upon the imperial person were carved alive and eaten by the assembled ministers on the main avenue south of the city.
##: '''Original:''' 太宗至城西，卫士执送绍。于是赐绍母子死，诛帐下阉官、宫人为内应者十数人，其先犯乘舆者，群臣于城南都街生脔割而食之。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 16 &quot;Biographies 4, The Seven Princes of Daowu&quot; (《魏书·卷十六·列传第四·道武七王》)
## 409 CE: Cannibalism as Punishment for Regicide, ''Zizhi Tongjian''
##: '''English''': Those who had first laid hands upon the imperial person [Tuoba Gui] were carved and eaten by the assembled ministers.
##: '''Original:''' 其先犯乘舆（拓跋珪）者，群臣脔食之。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']] (《资治通鉴》)

==南北朝==

#  431年: 赫连定遣其北平公韦代率众万人攻南安。城内大饥，人相食。（《北史·卷九十三·列传第八十一·僭伪附庸》㉕*）&lt;p&gt;赫连定遣其北平公韦代率众一万攻南安，城内大饥，人相食。（《魏书·卷九十九·列传第八十七·凉州牧张实等》㉕）&lt;/p&gt;&lt;p&gt;夏主（赫连定）击秦将姚献，败之；遂遣其叔父北平公韦伐帅众一万攻南安。城中大饥，人相食。（《资治通鉴》卷122）&lt;/p&gt;
#  宋人[[:w:劉敬叔|劉敬叔]]的《異苑》：“元嘉中，豫章胡家奴開邑王冢，青州人開[[:w:齊襄公|齊襄公]]冢，並得金鉤，而屍骸露在岩中儼然。茲亦未必有憑而然也，京房屍至義熙中猶完具，殭屍人肉堪為藥，軍士分割食之。”
#  441年，元嘉十八年：七月，拓跋焘遣军围酒泉。十月，城中饥，万余口皆饿死，（沮渠）天周杀妻以食战士；食尽，城乃陷，执天周至平城，杀之。（《宋书·卷九十八·列传第五十八·氐胡》㉕*）&lt;p&gt;酒泉城中食尽，万馀口皆饿死，沮渠天周杀妻以食战士。（《资治通鉴》卷123）&lt;/p&gt;
#   约450年，刘宋元嘉末：元嘉末，青州饥荒，人相食。（《南齐书· 卷二十八·列传第九·崔祖思等》㉕*）&lt;p&gt;元嘉末，青州饥荒，人相食。（刘）善明家有积粟，躬食饘粥，开仓以救，乡里多获全济，百姓呼其家田为续命田。（《南史·卷四十九·列传第三十九·庾杲之等》㉕）&lt;/p&gt;
#  453年，[[:w:宋文帝|宋文帝]]元嘉三十年（453年）：张超之闻兵入，遂走至合殿故基，正于御床之所，为乱兵所杀。割肠刳心，脔剖其肉，诸将生啖之，焚其头骨。（《宋书·卷九十九·列传第五十九·二凶》㉕*）&lt;p&gt;张超之闻兵入，遂至合殿故基，止于御床之所，为乱兵所杀，剖腹刳心，脔割其肉，诸将生啖之。焚其头骨。（《南史·卷十四·列传第四·宋宗室及诸王下》㉕）&lt;/p&gt;&lt;p&gt;张超之走至合殿御床之所。为军士所杀，刳肠割心，诸将脔其肉，生啖之。（《资治通鉴》卷127）&lt;/p&gt;
#   约454年: （刘）邕所至嗜食疮痂，以为味似鳆鱼。尝诣孟灵休，灵休先患灸疮，疮痂落床上，因取食之。灵休大惊。答曰：“性之所嗜。”灵休疮痂未落者，悉褫取以饴邕。邕既去，灵休与何勖书曰：“刘邕向顾见啖，遂举体流血。”南康国吏二百许人，不问有罪无罪，递互与鞭，鞭疮痂常以给膳。（《宋书·卷四十二·列传第二·刘穆之等》㉕*）&lt;p&gt; （刘）邕性嗜食疮痂，以为味似鳆鱼。尝诣孟灵休，灵休先患灸疮，痂落在床，邕取食之。灵休大惊，痂未落者，悉褫取饴邕。邕去，灵休与何勖书曰：“刘邕向顾见啖，遂举体流血。”南康国吏二百许人，不问有罪无罪，递与鞭，疮痂常以给膳。（《南史·卷十五·列传第五·刘穆之等》㉕）&lt;/p&gt;
#   465年: 前废帝（刘子业）狂悖无道，（王）义恭、（柳）元景谋欲废立，废帝率羽林兵于第害之，并其四子。断析义恭支体，分裂腹胃，挑取眼睛以蜜渍之，以为鬼目粽。（《南史· 卷十三·列传第三·宋宗室及诸王上》㉕*）&lt;p&gt;帝（南朝宋前废帝刘子业）自帅羽林兵讨（王）义恭，杀之，并其四子。断绝义恭支体，分裂肠胃，挑取眼睛，以蜜渍之，谓之“鬼目粽”。 （《资治通鉴》卷130）&lt;/p&gt;
#  498年:虏追军获（黄）瑶起，王肃募人脔食其肉。（《南齐书· 卷五十七·列传第三十八·魏虏》㉕*）&lt;p&gt;（王）琛弟肃、秉并奔魏，后得黄瑶起脔食之。（《南史·卷二十三·列传第十三·王诞等》㉕）&lt;/p&gt;&lt;p&gt;（黄）瑶起为魏所获，魏主以赐王肃，肃脔而食之。 （《资治通鉴》卷141）&lt;/p&gt;
#   499年，南齐永元元年: 永元元年，（陈）显达督平北将军崔慧景众军四万，围南乡堺马圈城，去襄阳三百里，攻之四十日。虏食尽，啖死人肉及树皮。（《南齐书· 卷二十六·列传第七·王敬则 陈显达》㉕*）&lt;p&gt;永元元年，（陈）显达督平北将军崔慧景众军四万，围南乡界马圈城，去襄阳三百里。攻之四十日，魏军食尽，啖死人肉及树皮。（《南史·卷四十五·列传第三十五·王敬则等》㉕）&lt;/p&gt;&lt;p&gt;陈显达与魏元英战，屡破之。攻马圈城四十日，城中食尽，啖死人肉及树皮。 （《资治通鉴》卷142）&lt;/p&gt;
#   502年：时东昏余党孙文明等……作乱，……（张）弘策踰垣匿于龙厩，遇贼见害。……官军捕文明斩于东市，张氏亲属脔食之。（《南史·卷五十六·列传第四十六·张弘策等》㉕*）
#   502年，梁天监元年：天监元年六月，元起至巴西，（侯）季连遣其将李奉伯拒战，见败。季连固守，元起围之。城中饿死者相枕，又从而相食。（《南史·卷十三·列传第三·宋宗室及诸王上》㉕*）&lt;p&gt;元起进屯西平，（侯）季连始婴城自守。时益州兵乱既久，人废耕农，内外苦饥，人多相食，道路断绝。季连计穷。（《南史·卷五十五·列传第四十五·王茂等》㉕）&lt;/p&gt;&lt;p&gt;时益部兵乱日久，民废耕农，内外苦饥，人多相食，道路断绝，季连计穷。（《梁书·卷十·列传第四·萧颖达等》㉕）&lt;/p&gt;
#  503年: 成都城中食尽，升米三千，人相食。（《资治通鉴》卷145）
#   约525年: 大将军萧宝夤西讨，德广为行台郎，募众而征，战捷，乃手刃仇人，啖其肝肺。（《北史·卷一百·序传第八十八》㉕*）
#   525年: 山胡刘蠡升自云圣术，胡人信之，咸相影附，旬日之间，逆徒还振。……先是官粟贷民。未及收聚，仍值寇乱。至是（汾州）城民大饥，人相食。贼知仓库空虚，攻围日甚，死者十三四。（裴）良以饥窘，因与城人奔赴西河。（《魏书·卷六十九·列传第五十七·崔休等》㉕*）
#  529年: 于是（元颢）斩（杨）昱下统帅三十七人，皆令蜀兵刳腹取心食之。（《北史·卷四十一·列传第二十九·杨播等》㉕*）&lt;p&gt;于是（元颢）斩（杨）昱下统帅三十七人，皆令蜀兵刳腹取心食之。（《魏书·卷五十八·列传第四十六·杨播》㉕）&lt;/p&gt;&lt;p&gt;于是（元颢）斩（杨）昱所部统帅三十七人，皆刳心而食之。 （《资治通鉴》卷153）&lt;/p&gt;
#   约532年：（北方）于时年凶，人多相食，昕勤恤人隐，多所全济。（《北史·卷二十四·列传第十二·崔逞等》㉕*）
#   约533年: 中大通四年，（梁武帝萧衍）特封（萧正德）临贺郡王。后为丹阳尹，坐所部多劫盗，复为有司所奏，去职。出为南兖州，在任苛刻，人不堪命。广陵沃壤，遂为之荒，至人相食啖。（《南史·卷五十一·列传第四十一·梁宗室上》㉕*）
#  536年: 是岁，关中大饥，人相食，死者十七八。（《北史·卷五·魏本纪第五》㉕*）&lt;p&gt; （西）魏关中大饥，人相食，死者什七八。  （《资治通鉴》卷157）&lt;/p&gt;
#  548年: 景食石头常平仓既尽，便掠居人，尔后米一升七八万钱，人相食，有食其子者。又筑土山，不限贵贱，昼夜不息，乱加殴棰，疲羸者因杀以填山，号哭之声动天地。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;石头常平诸仓既尽，（侯景）军中乏食；乃纵士卒掠夺民米及金帛子女。是后米一升直七八万钱，人相食，饿死者什五六。 （《资治通鉴》卷161）&lt;/p&gt;
#   548年: 鄱阳世子嗣、永安侯确、羊鸦仁、李迁仕、樊文皎率众度淮，攻破贼（侯景）东府城前栅，遂营于青溪水东。（侯）景遣其仪同宋子仙缘水西立栅以相拒。景食稍尽，人相食者十五六。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;景遣其仪同宋子仙顿南平王第，缘水西立栅相拒。景食稍尽，至是米斛数十万，人相食者十五六。（《梁书·卷五十六·列传第五十·侯景》㉕）&lt;/p&gt;
#   549年， [[:w:梁武帝|梁武帝]]太清三年：贼（侯景）之始至，（建邺）城中才得固守，平荡之事，期望援军。既而中外断绝，……军人屠马于殿省间鬻之，杂以人肉，食者必病。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;（萧）衍城内大饥，人相食，米一斗八十万，皆以人肉杂牛马而卖之。（《魏书·卷九十八·列传第八十六·岛夷萧道成等》㉕）&lt;/p&gt;&lt;p&gt;（梁）军人屠马于殿省间，杂以人肉，食者必病。 （《资治通鉴》卷162）&lt;/p&gt;
#  549年: 自（侯）景作乱，（建康）道路断绝，数月之间，人至相食，犹不免饿死，存者百无一二。贵戚、豪族皆自出采稆，填委沟壑，不可胜纪。  （《资治通鉴》卷162）
#   549年，梁太清三年：是月（七月），九江大饥，人相食十四五。（《梁书·卷四·本纪第四·简文帝》㉕*）&lt;p&gt;九江大饥，人相食者十四五。（《南史·卷八·梁本纪下第八》㉕）&lt;/p&gt;&lt;p&gt;是年，帝为侯景所幽，崩。七月，九江大饥，人相食十四五。（《隋书·卷二十一·志第十六·天文下》㉕）&lt;/p&gt;
#  550: 值梁室丧乱，（姚察）于金陵随二亲还乡里。时东土兵荒，人饥相食，告籴无处，察家口既多，并采野蔬自给。（《陈书· 卷二十七·列传第二十一·江总 姚察》㉕*）&lt;p&gt;自晋氏度江，三吴最为富庶，贡赋商旅，皆出其地。及侯景之乱，掠金帛既尽，乃掠人而食之，或卖于北境，遗民殆尽矣。  （《资治通鉴》卷163）&lt;/p&gt;
#   550年，梁大宝元年：自春迄夏，大饥，人相食，京师尤甚。（《梁书·卷四·本纪第四·简文帝》㉕*）&lt;p&gt;自春迄夏大旱，人相食，都下尤甚。（《南史·卷八·梁本纪下第八》㉕）&lt;/p&gt;
#   552年:（侯）景不能制，乃与腹心数十人单舸走，推堕二子于水，自沪渎入海。至壶豆洲，前太子舍人羊鲲杀之，送尸于王僧辩，传首西台，曝尸于建康市。百姓争取屠脍啖食，焚骨扬灰。（《梁书·卷五十六·列传第五十·侯景》㉕*）&lt;p&gt;及（侯）景死，僧辩截其二手送齐文宣，传首江陵，果以盐五斗置腹中，送于建康，暴之于市。百姓争取屠脍羹食皆尽，并溧阳主亦预食例。景焚骨扬灰，曾罹其祸者，乃以灰和酒饮之。（《南史·卷八十·列传第七十·贼臣》㉕）&lt;/p&gt;&lt;p&gt;既斩侯景，烹尸于建业市，百姓食之，至于肉尽龁骨，传首荆州，悬于都街。（《北齐书· 卷四十五·列传第三十七·文苑》㉕）&lt;/p&gt;&lt;p&gt;僧辩传（侯景）首江陵，截其手，使谢葳蕤送于齐；暴景尸于市，士民争取食之，并骨皆尽；溧阳公主亦预食焉。 （《资治通鉴》卷164）&lt;/p&gt;
#   552年: 王伟，陈留人。少有才学，景之表、启、书、檄，皆其所制。景既得志，规摹篡夺，皆伟之谋。及囚送江陵，烹于市，百姓有遭其毒者，并割炙食之。（《梁书·卷五十六·列传第五十·侯景》㉕*）
#  553年: （萧）圆照更无所言，唯云计误。并命绝食于狱，齿臂啖之，十三日死，天下闻而悲之。（《南史·卷五十三·列传第四十三·梁武帝诸子》㉕*）&lt;p&gt;上（梁元帝萧绎）并命（萧圆正）绝食于狱，至啮臂啖之，十三日而死，远近闻而悲之。 （《资治通鉴》卷165）&lt;/p&gt;
#  《南史》毗骞：“国法刑人，并于王前啖其肉。”“国内不受估客，往者亦杀而食之。”
#  554年: 五年春正月癸丑，帝（北齐文宣帝高洋）讨山胡大破之。男子十二已上皆斩，女子及幼弱以赏军。遂平石楼。石楼绝险，自魏代所不能至。于是远近山胡，莫不慑伏。是役也，有都督战伤，其什长路晖礼不能救，帝命刳其五藏，使九人分食之，肉及秽恶皆尽。自是始行威虐。（《北史·卷七·齐本纪中第七》㉕*）&lt;p&gt;有都督战伤，其什长路晖礼不能救，帝（北齐文宣帝高洋）命刳其五藏，令九人食之，肉及秽恶皆尽。（《资治通鉴》卷165）&lt;/p&gt;
#   555年: 众推（慕容）俨，遂遣镇郢城。……（侯）瑱、（任）约又并力围城。唯煮槐楮叶并纻根、水荭、葛、艾等及靴、皮带、筋角等食之。人死，即火别分食，唯留骸骨。俨犹信赏必罚，分甘同苦。自正月至六月，人无异志。（《北史·卷五十三·列传第四十一·万俟普等》㉕*）
#   约555年-560年: 自（天保）六年之后，帝（北齐文宣帝高洋）遂以功业自矜，恣行酷暴，昏狂酗醟，任情喜怒。为大镬、长锯、剉碓之属，并陈于庭，意有不快，则手自屠裂，或命左右脔啖，以逞其意。（《隋书·卷二十五·志第二十·刑法》㉕*）
#  流求国，居海岛，当建安郡东。水行五日而至。……国人好相攻击，……两军相当，勇者三五人出前跳噪，交言相骂，因相击射。如其不胜，一军皆走，遣人致谢，即共和解。收取斗死者聚食之，仍以髑髅将向王所，王则赐之以冠，便为队帅。……其南境风俗少异，人有死者，邑里共食之。（《北史·卷九十四·列传第八十二·高丽等》㉕*）&lt;p&gt;流求国，……南境风俗少异，人有死者，邑里共食之。（《隋书·卷八十一·列传第四十六·东夷》㉕）&lt;/p&gt;
#   獠者，盖南蛮之别种，自汉中达于邛、笮，川洞之间，所在皆有。……性同禽兽，至于忿怒，父子不相避，唯手有兵刃者先杀之。……若报怨相攻击，必杀而食之；（《北史·卷九十五·列传第八十三·蛮 獠 等》㉕*）
#   顿逊之外，大海洲中，又有毗骞国，去扶南八千里。……国法刑罪人，并于王前啖其肉。国内不受估客，有往者亦杀而啖之，是以商旅不敢至。（《梁书·卷五十四·列传第四十八·诸夷》㉕*）&lt;p&gt;又有毗骞国，去扶南八千里。……国法刑人，并于王前啖其肉。国内不受估客，有往者亦杀而啖之，是以商旅不敢至。（《南史· 卷七十八·列传第六十八·夷貊上》㉕）&lt;/p&gt;

==隋==
#   590年: 时江南州县又论言欲徙之入关，远近惊骇。饶州吴世华起兵为乱，生脔县令，啖其肉。（《北史·卷六十三·列传第五十一·周惠达等》㉕*）
#   隋文帝开皇年间（581-600年）：（杨武通）与周法尚讨嘉州叛獠，……贼知其孤军无援，倾部落而至。武通转斗数百里，为贼所拒，四面路绝。武通轻骑挑战，坠马，为贼所执，杀而啖之。（《北史·卷七十三·列传第六十一·梁士彦等》㉕*）&lt;p&gt;（杨）武通轻骑接战，坠马，为贼所执，杀而啖之。（《隋书·卷五十三·列传第十八·达奚长儒》㉕）&lt;/p&gt;
#   隋文帝开皇年间（581-600年）：郡中士女，号哭于路，诸郡惊骇，各奏其（王文同）事。帝闻大怒，遣使者违奚善意驰锁之（王文同），斩于河间，以谢百姓。仇人剖其棺，脔其肉啖之，斯须咸尽。（《北史·卷八十七·列传第七十五·酷吏》㉕*）&lt;p&gt;郡中士女号哭于路，诸郡惊骇，各奏其（王文同）事。帝闻而大怒，遣使者达奚善意驰锁之，斩于河间，以谢百姓，仇人剖其棺，脔其肉而啖之，斯须咸尽。（《隋书·卷七十四·列传第三十九·酷吏》㉕）&lt;/p&gt;
#   隋炀帝时代（604年-618年在位）中期：六军不息，百役繁兴；行者不归，居者失业；人饥相食，邑落为墟，上弗之恤也。（《北史·卷十二·隋本纪下第十二》㉕*）&lt;p&gt;六军不息，百役繁兴，行者不归，居者失业。人饥相食，邑落为墟，上不之恤也。（《隋书·卷四·帝纪第四·炀帝下》㉕）&lt;/p&gt;
#   613年: 及杨玄感反，帝（隋炀帝杨广）诛之，罪及九族。其尤重者，行轘裂枭首之刑。或磔而射之。命公卿已下，脔啖其肉。（《隋书·卷二十五·志第二十·刑法》㉕*）
#  614年:明年，（隋炀）帝复东征，高丽请和，遂送（斛斯）政。锁至京师以告庙，左翊卫大将军宇文述请变常法行刑，帝许之。以出金光门，缚之于柱，公卿百僚，并亲击射。脔其肉，多有啖者，然后烹焚，扬其骨灰。（《北史·卷四十九·列传第三十七·朱瑞等》㉕*）&lt;p&gt;（隋炀）帝复东征，高丽请降，求执送（斛斯）政。帝许之，遂锁政而还。至京师，以政告庙，左翊卫大将军字文述奏曰：“斛斯政之罪，天地所不容，人神所同忿。若同常刑，贼臣逆子何以惩肃？请变常法。”帝许之。于是将政出金光门，缚政于柱，公卿百僚并亲击射，脔割其肉，多有啖者。啖后烹煮，收其余骨，焚而扬之。（《隋书·卷七十·列传第三十五·杨玄感》㉕）&lt;/p&gt;&lt;p&gt;十一月，丙申，杀斛斯政于金光门外，如杨积善之法，仍烹其肉，使百官啖之，佞者或啖之至饱，收其馀骨，焚而扬之。 （《资治通鉴》卷182）&lt;/p&gt;
#   隋炀帝时代（604年-618年在位）后期：民外为盗贼所掠，内为郡县所赋，生计无遗；加之饥馑无食，民始采树皮叶，或捣穢为末，或煮土而食之，诸物皆尽，乃自相食。而官食犹充牣，吏皆畏法，莫敢振救。 （《资治通鉴》卷183）&lt;p&gt;相聚雚蒲，猬毛而起。大则跨州连郡，称帝称王；小则千百为群，攻城剽邑。流血成川泽，死人如乱麻；炊者不及析骸，食者不遑易子。（《北史·卷十二·隋本纪下第十二》㉕*）&lt;/p&gt;&lt;p&gt;俄而玄感肇黎阳之乱，匈奴有雁门之围，天子方弃中土，远之扬越。奸宄乘衅，强弱相陵，关梁闭而不通，皇舆往而不反。加之以师旅，因之以饥馑，流离道路，转死沟壑，十八九焉。于是相聚萑蒲，蝟毛而起，大则跨州连郡，称帝称王，小则千百为群，攻城剽邑，流血成川泽，死人如乱麻，炊者不及析骸，食者不遑易子。（《隋书·卷四·帝纪第四·炀帝下》㉕）&lt;/p&gt;&lt;p&gt;自燕赵跨于齐韩，江淮入于襄邓，东周洛邑之地，西秦陇山之右，僭伪交侵，盗贼充斥。宫观鞠为茂草，乡亭绝其烟火，人相啖食，十而四五。（《隋书·卷二十四·志第十九·食货》㉕）&lt;/p&gt;&lt;p&gt;是时百姓废业，屯集城堡，无以自给。然所在仓库，犹大充爨，吏皆惧法，莫肯赈救，由是益困。初皆剥树皮以食之，渐及于叶，皮叶皆尽，乃煮土或捣稿为末而食之。其后人乃相食。（《隋书·卷二十四·志第十九·食货》㉕）&lt;/p&gt;
#  616: 吏立木于市，悬其（张金称）头，张其手足，令仇家割食之；未死间，歌讴不辍。（《资治通鉴》卷183）
#  617年，大业十三年四月：（薛仁杲）所至多杀人，纳其妻妾。获庾信子立，怒其不降，磔于猛火之上，渐割以啖军士。（《旧唐书·卷五十五·列传第五·薛举等》㉕*）&lt;p&gt;（薛仁杲）尝得庾信子立，怒其不降，砾之火，渐割以啖士。（《新唐书·卷八十六·列传第十一 薛李二刘高徐》㉕）&lt;/p&gt;&lt;p&gt;（薛仁杲）尝获庾信子立，怒其不降，磔于火上，稍割以啖军士。”（《资治通鉴》卷183）&lt;/p&gt;
#  618: :（屈突）通引兵南遁，置（尧）君素领河东通守。……后颇得江都倾覆消息，又粮尽，男女相食，众心离骇。（《北史·卷八十五·列传第七十三·节义》㉕*）&lt;p&gt;时百姓苦隋日久，及逢义举，人有息肩之望。然君素善于统领，下不能叛。岁余，颇得外生口，城中微知江都倾覆。又粮食乏绝，人不聊生，男女相食，众心离骇。（《隋书·卷七十一·列传第三十六·诚节》㉕）&lt;/p&gt;&lt;p&gt;隋将尧君素守河东，上遣吕绍宗、韦义节、独孤怀恩相继攻之，俱不下。……久之，仓粟尽，人相食；（《资治通鉴》卷184）&lt;/p&gt;
#  618: （李轨）征兵筑台以候玉女，多所糜费，百姓患之。又属年饥，人相食，轨倾家赈之，私家罄尽，不能周遍。（谢统师等）乃诟珍曰：“百姓饿者自是弱人，勇壮之士终不肯困，国家仓粟须备不虞，岂可散之以供小弱？仆射苟悦人情，殊非国计。”轨以为然，由是士庶怨愤，多欲叛之。（《旧唐书·卷五十五·列传第五 薛举等》㉕*）&lt;p&gt;有胡巫妄曰：“上帝将遣玉女从天来。”（李轨）遂召兵筑台以候女，多所糜损。属荐饥，人相食，轨毁家赀赈之，不能给，议发仓粟，曹珍亦劝之。谢统师等故隋官，内不附，每引结群胡排其用事臣，因是欲离沮其众，乃廷诘珍曰：“百姓饥死皆弱不足事者，壮勇士终不肯困。且储禀以备不虞，岂宜妄散惠孱小乎？仆射苟附下，非国计。”轨曰：“善。”乃闭粟。下益怨，多欲叛去。（《新唐书·卷八十六·列传第十一·薛李二刘高徐》㉕） &lt;/p&gt;&lt;p&gt;有胡巫谓（李）轨曰：“上帝当遣玉女自天而降。”轨信之，发民筑台以候玉女，劳费甚广。河右饥，人相食，轨倾家财以赈之；不足，欲发仓粟，召群臣议之。曹珍等皆曰：“国以民为本，岂可爱仓粟而坐视其死乎！”谢统师等皆故隋官，心终不服，密与群胡为党，排轨故人，乃诟珍曰：“百姓饿者自是羸弱，勇壮之士终不至此。国家仓粟以备不虞，岂可散之以饲羸弱！仆射苟悦人情，不为国计，非忠臣也。”轨以为然，由是士民离怨。 （《资治通鉴》卷186）&lt;/p&gt;
#  619年：（朱）粲所克州县，皆发其藏粟以充食，迁徙无常，去辄焚余赀，毁城郭，又不务稼穑，以劫掠为业。于是百姓大馁，死者如积，人多相食。军中罄竭，无所虏掠，乃取婴儿蒸而啖之，因令军士曰：“食之美者，宁过于人肉乎！但令他国有人，我何所虑？”即勒所部，有略得妇人小儿皆烹之，分给军士，乃税诸城堡，取小弱男女以益兵粮。隋著作佐郎陆从典、通事舍人颜愍楚因谴左迁，并在南阳，粲悉引之为宾客，后遭饥馁，合家为贼所啖。（《旧唐书·卷五十六·列传第六·萧铣等》㉕*）&lt;p&gt;粲所克州县皆发藏粟以食，迁徙无常，去辄燔廥聚，毁城郭，不务稼穑，专以劫为资。于是人大馁，死者系路，其军亦匮，乃掠小儿烝食之。戒其徒曰：“味之珍宁有加人者？弟使佗国有人，我恤无储哉！”勒所部略妇人孺儿分烹之，又税诸城细弱以益粮。隋著作佐郎陆从典、通事舍人颜愍楚谪南阳，粲初引为宾客，后尽食两家。俄而诸城惧，皆逃散。（《新唐书·卷八十七·列传第十二·萧辅沈李梁》㉕）&lt;/p&gt;&lt;p&gt;朱粲有众二十万，剽掠汉、淮之间，迁徙无常，攻破州县，食其积粟未尽，复他适，将去，悉焚其余资；又不务稼穑，民馁死者如积。粲无可复掠，军中乏食，乃教士卒烹妇人、婴儿啖之，曰：“肉之美者无过于人，但使他国有人，何忧于馁！”隋著作佐郎陆从典、通事舍人颜愍楚，谪官在南阳，粲初引为宾客，其后无食，阖家皆为所啖。愍楚，之推之子也。又税诸城堡细弱以供军食，诸城堡相帅叛之。”（《资治通鉴》）&lt;/p&gt;&lt;p&gt;“隋末荒亂，狂賊[[:w:朱粲|朱粲]]起於襄、鄧間，歲飢，米斛萬錢，亦無得處，人民相食。粲乃驅男女小大仰一大銅鐘，可二百石，煮人肉以矮賊。生靈殲於此矣。”，朱粲竟說：“食之美者，寧過於人肉乎！”（唐·[[:w:張鷟|張鷟]]《朝野僉載》）&lt;/p&gt;
#  619年: （段）确醉，戏（朱）粲曰：“君脍人多矣，若为味？”粲曰：“啖嗜酒人，正似糟豚。”确悸，骂曰：“狂贼，归朝乃一奴耳，复得噬人乎？”粲惧，收确于坐，并从者数十悉饔之，以飨左右。遂屠菊潭，奔王世充，署龙骧大将军。东都平，斩洛水上。士庶竞掷瓦砾击其尸，须臾若冢。（《新唐书·卷八十七·列传第十二·萧辅沈李梁》㉕*）&lt;p&gt;（段确）乘醉侮（朱）粲曰：“闻卿好啖人，人作何味？”粲曰：“啖醉人正如糟藏彘肉。”确怒，骂曰：“狂贼入朝，为一头奴耳，复得啖人乎！”粲于座收确及从者数十人，悉烹之，以啖左右。（《资治通鉴》卷187）&lt;/p&gt;
#  隋末的[[:w:诸葛昂|诸葛昂]]與[[:w:高瓒|高瓒]]嗜食人肉。高瓒將双胞胎小孩杀掉，頭顱、手和腳分別裝在盤子裏，做成“双子宴”，與诸葛昂一起享用；诸葛昂则把自己的爱妾蒸熟，擺成盤腿打坐的姿勢，臉上重新塗好脂粉，諸葛昂親手撕她大腿上的肉請高瓒吃。（《[[:w:唐人说荟|唐人说荟]]》卷五，引张骞《耳目记》）

==唐==
安史之乱期间，张巡固守城池，城中人相食，张巡杀妾以飨将士，对于张巡以食人为代价的守土之功是否应该奖励，出现了一次伦理学的辩论，历代不息，《柏杨白话版资治通鉴》收集了若干历史上争论的意见。

黄巢之乱的时候，几支反叛军队成规模地常规性地以人为食，黄巢军“掠人为粮，生投于碓硙，并骨食之，号给粮之处曰‘舂磨寨’”，秦宗权军“啖人为储，军士四出，则盐尸而从”，李罕之军“不耕稼，专以剽掠为资，啖人为粮”。真是惨烈之甚。

唐朝陈藏器写的《本草拾遗》写人肉可以治病，这应该不是他的发明，而只是民间认知的一种总结，可能只是太多不得已的饥荒食人造成一种认知扭曲，但又反过来理性化了食人，到宋朝的时候，割肉疗亲开始出现。

#   621年，[[:w:唐高祖|唐高祖]]武德四年：（王）世充屯兵不散，仓粟日尽，城中人相食。或握土置瓮中，用水淘汰，沙石沉下，取其上浮泥，投以米屑，作饼饵而食之，人皆体肿而脚弱，枕倚于道路。其尚书郎卢君业、郭子高等皆死于沟壑。（《旧唐书·卷五十四·列传第四 王世充 窦建德》㉕*）&lt;p&gt;王（李世民）傅城，堑而守之。（王）世充粮且尽，人相食，至以水汨泥去砾，取浮土糅米屑为饼。民病肿股弱，相藉倚道上，其尚书郎卢君业、郭子高等皆饿死。御史大夫郑颋丐为浮屠，世充恶其言，杀之。（《新唐书·卷八十五·列传第十 王窦》㉕）&lt;/p&gt;
#621年: （单雄信）临将就戮，（李世）勣对之号恸，割股肉以啖之，曰：“生死永诀，此肉同归于土矣。”（《旧唐书·卷六十七·列传第十七·李靖等》㉕*）&lt;p&gt;（李世勣）乃割股肉以啖（单）雄信，曰：“使此肉随兄为土，庶几犹不负昔誓也！”（《资治通鉴》卷189）&lt;/p&gt;
#   627年: （王）君操密袖白刃刺杀之（杀父仇人李君则），刳腹取其心肝，啖食立尽，诣刺史具自陈告。（《旧唐书·卷一百八十八·列传第一百三十八·孝友》㉕*）
#   643年，[[:w:唐太宗|唐太宗]]贞观十七年: 贞观末，（刘兰）以谋反腰斩。右骁卫大将军丘行恭探其心肝而食之，太宗闻而召行恭让之曰：“典刑自有常科，何至于此！必若食逆者心肝而为忠孝，则刘兰之心为太子诸王所食，岂至卿邪？”行恭无以答。（《旧唐书·卷六十九·列传第十九·侯君集等》㉕*）&lt;p&gt;鄠尉[[:w:游文芝|游文芝]]告代州都督[[:w:劉蘭成|劉蘭成]]谋反，戊申，兰成坐[[:w:腰斩|腰斩]]。右武候将军[[:w:丘行恭|丘行恭]]，探兰成心肝食之。上（唐太宗）闻而让之曰：兰成谋反，国有常刑，何至如此！若以为忠孝，则太子诸王先食之矣，岂至卿耶？行恭惭而拜谢。（《资治通鉴》卷196）&lt;/p&gt;
#   约650年：周智寿者，雍州同官人。其父永徽初被族人安吉所害。智寿及弟智爽乃候安吉于途，击杀之。兄弟相率归罪于县，争为谋首，官司经数年不能决。乡人或证智爽先谋，竟伏诛。临刑神色自若，顾谓市人曰：“父仇已报，死亦何恨！”智寿顿绝衢路，流血遍体。又收智爽尸，舐取智爽血，食之皆尽，见者莫不伤焉。（《旧唐书·卷一百八十八·列传第一百三十八·孝友》㉕*）
#   662年: （郑）仁泰选骑万四千卷甲驰，绝大漠，至仙萼河，不见虏，粮尽还。人饥相食，比入塞，余兵才二十之一。（《新唐书·卷一百一十一·列传第三十六·郭二张三王苏薛程唐》㉕*)&lt;p&gt;（郑）仁泰将轻骑万四千，倍道赴之，遂逾大碛，至仙萼河，不见虏，粮尽而还。值大雪，士卒饥冻，弃捐甲兵，杀马食之，马尽，人自相食，比入塞，馀兵才八百人。（《资治通鉴》卷200）&lt;/p&gt;
#   682年，[[:w:唐高宗|唐高宗]]永淳元年：关中先水后早蝗，继以疾疫，米斗四百，两京间死者相枕于路，人相食。”（《资治通鉴》卷203）&lt;p&gt;六月，关中初雨，麦苗涝损，后旱，京兆、岐、陇螟蝗食苗并尽，加以民多疫疠，死者枕藉于路，诏所在官司埋瘗。京师人相食，寇盗纵横。（《旧唐书·卷五本纪第五·高宗下》㉕*）&lt;/p&gt;&lt;p&gt;永淳中，为雍州长史。时关中大饥，人相食，盗贼纵横。（《旧唐书·卷七十五·列传第二十五·苏世长等》㉕）&lt;/p&gt;&lt;p&gt;是月，大蝗，人相食。（《新唐书·卷三·本纪第三·高宗》㉕）&lt;/p&gt;&lt;p&gt;永淳元年，关中及山南州二十六饥，京师人相食。（《新唐书·卷三十五·志第二十五》㉕）&lt;/p&gt;&lt;p&gt;（良嗣）徙雍州。时关内饥，人相食，良嗣政上严，每盗发，三日内必擒，号称神明。（《新唐书·卷一百三·列传第二十八·苏世长等》㉕）&lt;/p&gt;
#   约684年: 王友贞，怀州河内人也。父知敬，则天时麟台少监，以工书知名。友贞弱冠时，母病笃，医言唯啖人肉乃差。友贞独念无可求治，乃割股肉以饴亲，母病寻差。则天闻之，令就其家验问，特加旌表。（《旧唐书·卷一百九十二·列传第一百四十二·隐逸》㉕*）
#   [[:w:武則天|武則天]]時期，杭州臨安縣尉薛震好吃人肉，“有債主及奴詣臨安，于客舍，遂飲之醉。殺而臠之，以水銀和煎，并骨消盡。后又欲食其婦，婦覺而遁。縣令詰得其情，申州，錄事奏，奉敕杖殺之。”（《[[:w:朝野僉載|朝野僉載]]》）
#   武則天時期，“周岭南首陳元光設客，令一袍褲行酒。光怒，令曳出，遂殺之。須臾爛煮，以食諸客。后呈其二手，客懼，攫喉而吐。”（出《摭言》。明抄本作出《朝野僉載》）
#   697年: 丁卯，（李）昭德、（来）俊臣同弃市，时人无不痛昭德而快俊臣。仇家争啖俊臣之肉，斯须而尽，抉眼剥面，披腹出心，腾蹋成泥。（《资治通鉴》卷206）
#   张鷟《[[s:朝野僉載_(四庫全書本)/卷2|朝野佥载]]》卷二：“后诛易之昌宗等，百姓脔割其肉，肥白如猪肪，煎炙而食。”
#   唐玄宗開元中葉人[[:w:陳藏器|陳藏器]]（713年－741年）《[[:w:本草拾遺|本草拾遺]]》寫吃人肉可以治病。
#   739年: 内给事牛仙童使幽州，受张守珪厚赂。玄宗怒，命思勖杀之。思勖缚架之数日，乃探取其心，截去手足，割肉而啖之，其残酷如此。（《旧唐书·卷一百八十四·列传第一百三十四·宦官》㉕*）&lt;p&gt; 内给事牛仙童纳张守珪赂，诏付思勖杀之。思勖缚于格，箠惨不可胜，乃探心，截手足，剔肉以食，肉尽乃得死。（《新唐书·卷二百七·列传第一百三十二·宦者上》㉕）&lt;/p&gt;&lt;p&gt;739年: 上（唐玄宗李隆基）怒，甲戌，命杨思勖杖杀之（牛仙童）。思勖缚格，杖之数百，刳取其心，割其肉啖之。（《资治通鉴》卷214）&lt;/p&gt;
#   757年: （鲁）炅城中食尽，煮牛皮筋角而食之，米斗至四五十千，有价无米，鼠一头至四百文，饿死者相枕藉。……炅在围中一年，救兵不至，昼夜苦战，人相食。（《旧唐书·卷一百一十四·列传第六十四·鲁炅等》㉕*）&lt;p&gt;（鲁）炅被围凡一年，昼夜战，人至相食，卒无救。（《新唐书·卷一百四十七·列传第七十二·三王鲁辛冯三李曲二卢》㉕）&lt;/p&gt;
#   757年: 尹子奇攻围（睢阳）既久，城中粮尽，易子而食，析骸而爨，人心危恐，虑将有变。（张）巡乃出其妾，对三军杀之，以飨军士。曰：“诸公为国家戮力守城，一心无二，经年乏食，忠义不衰。巡不能自割肌肤，以啖将士，岂可惜此妇，坐视危迫。”将士皆泣下，不忍食，巡强令食之。乃括城中妇人；既尽，以男夫老小继之，所食人口二三万，人心终不离变。（《旧唐书·卷一百八十七下·列传第一百三十七·忠义下》㉕*）&lt;p&gt;（张）巡士多饿死，存者皆痍伤气乏。巡出爱妾曰：“诸君经年乏食，而忠义不少衰，吾恨不割肌以啖众，宁惜一妾而坐视士饥？”乃杀以大飨，坐者皆泣。巡强令食之，远亦杀奴僮以哺卒，至罗雀掘鼠，煮铠弩以食。……被围久，初杀马食，既尽，而及妇人老弱凡食三万口。人知将死，而莫有畔者。城破，遣民止四百而已。 （《新唐书·卷一百九十二·列传第一百一十七·忠义中》㉕） &lt;/p&gt;&lt;/p&gt;（张巡守睢阳，）茶纸既尽，遂食马；马尽，罗雀掘鼠；雀鼠又尽，巡出爱妾，杀以食士，远亦杀其奴；然后括城中妇人食之；既尽，继以男子老弱。人知必死，莫有叛者，所馀才四百人。 （《资治通鉴》卷220）&lt;/p&gt;
#   758年: 明年，改乾元元年，伪德州刺史王暕、贝州刺史宇文宽等皆归顺，河北诸军各以城守累月，贼使蔡希德、安太清急击，复陷于贼，虏之以归，脔食其肉。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕*）
#   759年: 二年正月，史思明自率范阳精卒复陷魏州，乃伪称燕王。王师虽众，军无统帅，进退无所承禀，自冬徂春，竟未破贼，但引漳水以灌其城，城中食尽，易子而食。（《旧唐书·卷一百二十·列传第七十·郭子仪等》㉕*）&lt;p&gt; （安）庆绪自十月被围至二月，城中人相食，米斗钱七万余，鼠一头直数千，马食隤墙麦鞬及马粪濯而饲之。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕）&lt;/p&gt;&lt;p&gt;（郭子仪军）连营进围相州，引漳水灌城，漫二时，不能破。城中粮尽，人相食。庆绪求救于史思明。（《新唐书·卷一百三十七·列传第六十二·郭子仪》㉕）&lt;/p&gt;&lt;p&gt; 乾元元年秋九月，帝诏郭子仪率九节度兵凡二十万讨庆绪，攻卫州，……王师围已固，筑浚城隍三周，决安阳水灌城。城中栈而处，粮尽，易口以食，米斗钱七万余，一鼠钱数千，屑松饲马，隤墙取麦秸，濯粪取刍，城中欲降不得。（《新唐书·卷二百二十五上·列传第一百五十上·逆臣上》㉕）&lt;/p&gt;
#   760年: 有纳赂于上言求官者，（吕）諲补之蓝田尉。五月，上言事泄笞死，以其肉令从官食之，諲坐贬太子宾客。（《旧唐书·卷一百八十五下·列传第一百三十五·良吏下》㉕*）
#   760年:   三品钱行浸久，属岁荒，米斗至七千钱，人相食。 （《资治通鉴》卷221）
#   760年: 时大雾，自四月雨至闰月末不止。米价翔贵，人相食，饿死者委骸于路。（《旧唐书·卷十·本纪第十·肃宗》㉕*）&lt;p&gt; 是时自四月初大雾大雨，至闰四月末方止。是月，逆贼史思明再陷东都，米价踊贵，斗至八百文，人相食，殍尸蔽地。（《旧唐书·卷三十六·志第十六·天文下》㉕） &lt;/p&gt;&lt;p&gt;乾元三年闰四月，大雾，大雨月余。是月，史思明再陷东都，京师米斗八百文，人相食，殍骸蔽地。（《旧唐书·卷三十七·志第十七·五行》㉕）&lt;/p&gt;
#   761年: 时洛阳四面数百里，人相食，州县为墟。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕*）&lt;p&gt; 朝义虚怀礼下，事皆决大臣，然无经略才。当此时，洛阳诸郡人相食，城邑榛墟，（《新唐书·卷二百二十五上·列传第一百五十上·逆臣上》㉕）&lt;/p&gt;
#  [[:w:唐代宗|唐代宗]]廣德元年（763年），江東大疫，“死者過半”，[[:w:獨孤及|獨孤及]]描述這次的災難：“辛丑歲（762年），大旱，三吳飢甚，人相食。明年大疫，死者十七八，城郭邑居為之空虛，而存者無食，亡者無棺殯悲哀之送。大抵雖其父母妻子也啖其肉，而棄其骸於田野，由是道路積骨相支撐枕藉者彌二千里，春秋以來不書。”（《吊道殣文》）&lt;p&gt;江、淮大饥，人相食。（《资治通鉴》卷222）&lt;/p&gt;
#   [[:w:白居易|白居易]]（772年－846年）寫《輕肥》一詩有“是歲江南旱，衢州人食人。”
#   [[:w:張茂昭|張茂昭]]為節鎮，頻吃人肉，及除統軍，到京。班中有人問曰：聞尚書在鎮好人肉，虛實？” 昭笑曰：“人肉腥而且肕，爭堪吃。”（《盧氏雜記》）
#   766年: 监军张志斌自陕入奏，（周）智光馆给礼慢，志斌责其不肃。智光大怒曰：“仆固怀恩岂有反状！皆由尔鼠辈作福作威，惧死不敢入朝。我本不反，今为尔作之。”因叱下斩之，脔其肉以饲从者。（《旧唐书·卷一百一十四·列传第六十四·鲁炅等》㉕*）&lt;p&gt;（周智光）叱下斩之（张志斌），脔食其肉。（《资治通鉴》卷224）&lt;/p&gt;
#   775年:承嗣既令（田）廷玠（或作田庭玠）守沧州，而（李）宝臣、朱滔兵攻击，欲兼其土宇。廷玠婴城固守，连年受敌，兵尽食竭，人易子而食，卒无叛者，卒能保全城守。（《旧唐书·卷一百四十一·列传第九十一·田承嗣等》㉕*）
#   796年: 军士又呼曰：“仓官刘叔何给纳有奸。”杀而食之。（《资治通鉴》卷235）
#   799年: 是日，汴州军乱，杀陆长源及节度判官孟叔度、丘颖，军人脔而食之。（《旧唐书·卷十三·本纪第十三·德宗下》㉕*）&lt;p&gt;兵士怨怒滋甚，乃执长源及叔度等脔而食之，斯须骨肉糜散。（《旧唐书·卷一百四十五·列传第九十五·刘玄佐等》㉕）&lt;/p&gt;&lt;p&gt;才八日，军乱，杀长源及叔度等，食其肉，放兵大掠。（《新唐书·卷一百五十一·列传第七十六·关董袁赵窦》㉕）&lt;/p&gt;&lt;p&gt;是日，军士作乱，杀（陆）长源、（孟）叔度，脔食之，立尽。（《资治通鉴》卷235）&lt;/p&gt;
#   803年: 盐夏节度判官崔文先权知盐州，为政苛刻。冬，闰十月，庚戌，部将李庭俊作乱，杀而脔食之。（《资治通鉴》卷236）
#   807年: 锜不自安，亦请入朝，乃拜锜左仆射。锜乃署判官王澹为留后。既而迁延发期，澹与中使频喻之，不悦，遂讽将士以给冬衣日杀澹而食之。监军使闻乱，遣衙将赵锜慰喻，又脔食之。（《旧唐书·卷一百一十二·列传第六十二·李暠等》㉕*）&lt;p&gt;会使者召锜，称疾，留后王澹为具行，锜怒，阴教士脔食之，即胁使者为众奏天子，幸得留。（《新唐书·卷一百八十一·列传第一百六·陈夷行等》㉕）&lt;/p&gt;&lt;p&gt;807: （李）锜严兵坐幄中，（王）澹与敕使入谒，有军士数百噪于庭曰：“王澹何人，擅主军务！”曳下，脔食之；大将赵琦出慰止，又脔食之（《资治通鉴》卷237）&lt;/p&gt;
#   817年: 蔡将有李端者，过溵河降重胤。其妻为贼束缚于树，脔食至死，将绝，犹呼其夫曰：“善事乌仆射。”（《旧唐书·卷一百六十一·列传第一百一十一·李光进等》㉕*）&lt;p&gt;李湍妻。湍，吴元济之军人也。元和中，淮南未平，湍心怀向顺，乃急渡溵河，东降乌重胤。其妻遂为贼束缚在树，脔而食之，至死，叫其夫曰：“善事乌仆射。”观者义之。至是，重胤以其事请列史册。十三年，宪宗下诏从之。（《旧唐书·卷一百九十四上·列传第一百四十四上·突厥上》㉕）&lt;/p&gt;&lt;p&gt;李湍妻某氏。湍籍吴元济军，元和中，自拔归鸟重胤，妻为贼缚而脔食之，将死，犹号湍曰：“善事鸟仆射！”观者叹泣。重胤请以其事属史官，诏可。（《新唐书·卷二百五·列传第一百三十·列女》㉕）&lt;/p&gt;
#   822年: （王）播至淮南，属岁旱俭，人相啖食，课最不充，设法掊敛，比屋嗟怨。（《旧唐书·卷一百六十四·列传第一百一十四·王播等》㉕*）&lt;p&gt; 是时，南方旱歉，人相食，（王）播掊敛不少衰，民皆怨之。（《新唐书·卷一百六十七·列传第九十二·白裴崔韦二李皇甫王》㉕）&lt;/p&gt;
#   829年: 属岁旱俭，人至相食，楚均富赡贫，而无流亡者。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）
#   832年:（李）听先遣亲吏至徐州慰劳将士，苍头不欲听复来，说军士杀其亲吏，脔食之。（《资治通鉴》卷244）
#   约841年: （杜牧）作《罪言》。其辞曰：……. 山东叛且三五世，后生所见言语举止，无非叛也，以为事理正当如此，沉酣入骨髓，无以为非者，至有围急食尽，啖尸以战。以此为俗，岂可与决一胜一负哉？（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕*）
#   868年: 其年冬，庞勋杀崔彦曾，据徐州，聚众六七万。徐无兵食，乃分遣贼帅攻剽淮南诸郡，滁、和、楚、寿继陷。谷食既尽，淮南之民多为贼所啖。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）&lt;p&gt; 勋还，果盗徐州，其众六七万。徐乏食，分兵攻滁、和、楚、寿，陷之，粮尽，啖人以饱。（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕）&lt;/p&gt;
#   868年: 一日，贼军乘间，步骑径入湘垒，淮卒五千人皆被生絷送徐州，为贼蒸而食之。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）&lt;/p&gt;&lt;p&gt;湘乃彻警释械，日与勋众欢言。后贼乘间直袭湘垒，悉俘而食之，醢湘及监军郗厚本。（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕）&lt;/p&gt;
#   868年: 庞勋又令将刘贽攻濠州，陷之，囚刺史卢望回于回车馆，望回郁愤而死，仆妾数人皆为贼蒸而食之。（《旧唐书·卷十九上·本纪第十九上·懿宗》㉕*）
#   869年: 吴迥守濠州，粮尽食人，驱女孺运薪塞隍，并填之，整旅而行，马士举斩以献。（《新唐书·卷一百四十八·列传第七十三·令狐张康李刘田王牛史》㉕*）&lt;p&gt;马举攻濠州，自夏及冬不克，城中粮尽，杀人而食之（《资治通鉴》卷251）&lt;/p&gt;
#   876年：李廷节妻崔。乾符中，廷节为郏城尉。王仙芝攻汝州，廷节被执。贼见崔妹美，将妻之，诟曰：“我，士人妻，死亡有命，奈何受贼污？”贼怒，刳其心食之。（《新唐书·卷二百五·列传第一百三十·列女》㉕*）
#   878年: （李）尽忠械文楚等五人送斗鸡台下，（李）克用令军士玼食之，以骑践其骸。（《资治通鉴》卷253）
#   881年，[[:w:唐僖宗|唐僖宗]]廣明二年：（[[:w:黃巢|黃巢]]攻佔長安，）時京畿百姓皆寨于山谷，累年費耕耘，賊坐空城，賦輸無如，谷食騰踴，米斗三十錢，官軍皆執山寨百姓，蠰于賊為食，人獲數十萬”（《[[:w:舊唐書|舊唐書]]·卷二百下·列传第一百五十·朱泚 黄巢 秦宗权》㉕*）&lt;p&gt; 二年春正月甲辰朔，天下勤王之师，云会京畿，京师食尽。贼食树皮，以金玉买人于行营之师，人获数百万。山谷避乱百姓，多为诸军之所执卖。（《旧唐书·卷十九下·本纪第十九下 僖宗》㉕）&lt;/p&gt;&lt;p&gt;于时畿民栅山谷自保，不得耕，米斗钱三十千，屑树皮以食，有执栅民鬻贼以为粮，人获数十万钱。（《新唐书·卷二百二十五下·列传第一百五十下·逆臣下》㉕）&lt;/p&gt;&lt;p&gt;民避乱皆入深山筑栅自保，农事俱废，长安城中斗米直三十缗。贼（黄巢）卖人于官军以为粮，官军或执山栅之民鬻之，人直数百缗，以肥瘠论价。（《资治通鉴》卷254）&lt;/p&gt;
#  883年，唐僖宗中和三年883年：时黄巢与宗权合从，纵兵四掠，远近皆罹其酷。时仍岁大饥，民无积聚，贼俘人为食，其炮炙处谓之“舂磨寨”，白骨山积，丧乱之极，无甚于斯。（《旧唐书·卷十九下·本纪第十九下 僖宗》㉕*）&lt;p&gt;贼（黄巢）围陈郡百日，关东仍岁无耕稼，人饿倚墙壁间，贼俘人而食，日杀数千。贼有舂磨砦，为巨碓数百，生纳人于臼碎之，合骨而食，其流毒若是。（《旧唐书·卷二百下·列传第一百五十 朱泚 黄巢 秦宗权》㉕）&lt;/p&gt;&lt;p&gt;巢已东，使孟楷攻蔡州。节度使秦宗权迎战，大败，即臣贼，与连和。楷击陈州，败死，巢自围之，略邓、许、孟、洛，东入徐、兖数十州。人大饥，倚死墙堑，贼俘以食，日数千人，乃办列百巨碓，糜骨皮于臼，并啖之。（《新唐书·卷二百二十五下·列传第一百五十下 逆臣下》㉕）&lt;/p&gt;&lt;p&gt;是时，陈州四面，贼寨相望，驱掳编氓，杀以充食，号为“舂磨寨”。（《旧五代史·卷一（梁书）·太祖纪一》㉕）&lt;/p&gt;&lt;p&gt;秦宗权以蔡州附巢，巢势甚盛，乃悉众围犨，置舂磨，糜人之肉以为食。（《新五代史·卷四十二·杂传第三十·朱宣等》㉕）&lt;/p&gt;&lt;p&gt;时民间无积聚，贼（黄巢）掠人为粮，生投于碓硙，并骨食之，号给粮之处曰“舂磨寨”。纵兵四掠，自河南、许、汝、唐、邓、孟、郑、汴、曹、濮、徐、兖等数十州，咸被其毒。 （《资治通鉴》卷255）&lt;/p&gt;
#   884年: （秦宗权）所至屠翦焚荡，殆无孑遗。其残暴又甚于巢，军行未始转粮，车载盐尸以从。北至卫、滑，西及关辅，东尽青、齐，南出江、淮，州镇存者仅保一城，极目千里，无复烟火。（《资治通鉴》卷256）&lt;p&gt; 巢贼虽平，而宗权之凶徒大集，西至金、商、陕、虢，南极荆、襄，东过淮甸，北侵徐、兖、汴、郑，幅员数十州。五六年间，民无耕织，千室之邑，不存一二，岁既凶荒，皆脍人而食，丧乱之酷，未之前闻。（《旧唐书·卷二十上·本纪第二十上·昭宗》㉕*）&lt;/p&gt;&lt;p&gt;（秦宗权）贼首皆慓锐惨毒，所至屠残人物，燔烧郡邑。西至关内，东极青、齐，南出江淮，北至卫滑，鱼烂鸟散，人烟断绝，荆榛蔽野。贼既乏食，啖人为储，军士四出，则盐尸而从。（《旧唐书·卷二百下·列传第一百五十·朱泚 黄巢 秦宗权》㉕）&lt;/p&gt;&lt;p&gt; 中和二年，关内大饥。四年，关内大饥，人相食。（《新唐书·卷三十五·志第二十五 稼穑不成》㉕）&lt;/p&gt;&lt;p&gt;中和四年，江南大旱，饥，人相食。（《新唐书·卷三十五·志第二十五·常旸》㉕）&lt;/p&gt;
#   886年: 荆南、襄阳仍岁蝗旱，米斗三十千，人多相食。（《旧唐书·卷十九下·本纪第十九下·僖宗》㉕*）&lt;p&gt; 光启二年二月，荆、襄大饥，米斗三千钱，人相食。（《新唐书·卷三十五·志第二十五·稼穑不成》㉕）&lt;/p&gt;&lt;p&gt;二年，荆、襄蝗、米斗钱三千，人相食；（《新唐书·卷三十六·志第二十六·五行三》㉕）&lt;/p&gt;
#   886年: （张）瑰固垒二岁，樵苏皆尽，米斗钱四十千，计抔而食，号为“通肠”。疫死者，争啖其尸，县首于户以备馔。（《新唐书·卷一百八十六·列传第一百一十一 ·周王邓陈齐赵二杨顾》㉕*）
#   887年: 戊午，秦彦遣毕师铎、秦稠将兵八千出（扬州）城，西击杨行密。稠败死，士卒死者什七八。城中乏食，樵采路绝，宣州军始食之。（《资治通鉴》卷257）&lt;p&gt;五月，寿州刺史杨行密率兵攻（秦）彦，……重围半年，（扬州）城中刍粮并尽，草根木实、市肆药物、皮囊革带，食之亦尽。外军掠人而卖，人五十千。死者十六七，纵存者鬼形鸟面，气息奄然。（《旧唐书·卷一百八十二·列传第一百三十二·王重荣等》㉕*）&lt;/p&gt;&lt;p&gt;杨行密围扬州，毕师铎厚赍宝币，啖（杜）雄连和。雄率军浮海屯东塘。是时扬州围久，皮囊革带食无余，军中杀人代粮，才千钱。（《新唐书·卷一百九十·列传第一百一十五·三刘成杜钟张王》㉕）&lt;/p&gt;&lt;p&gt;是时，城中仓廪空虚，饥民相杀而食，其夫妇、父子自相牵，就屠卖之，屠者刲剔如羊豕。（《新五代史·卷六十一·吴世家第一》㉕）&lt;/p&gt;
#   887年: （高）骈家属并在道院，秦彦供给甚薄，薪蒸亦阙。奴仆彻延和阁栏槛煮革带食之，互相篡啖。（《旧唐书·卷一百八十二·列传第一百三十二·王重荣等》㉕*）&lt;p&gt;高骈在道院，秦彦供给甚薄，左右无食，至然木像、煮革带食之，有相啖者。（《资治通鉴》卷257）&lt;/p&gt;
#   887年，光启三年：（杨）行密攻围（广陵）弥急，城中食尽，米斗四十千，居人相啖略尽。十月，城陷，秦、毕走东塘，行密入广陵，辇外寨之粟以食饥民，即日米价减至三千。（《旧五代史·卷一百三十四·僭伪列传一》㉕*）&lt;p&gt;[[:w:杨行密|杨行密]]围广陵且半年，秦彦、毕师铎大小数十战多不利，城中无食，料值钱五十缗，草根木实皆尽，以堇泥为饼食之，饿死者大半。宣州军掠人诣肆卖之，驱缚屠割如羊豕，讫无一声，流血满于坊市。彦、师铎无如之何，颦蹙而已。（《资治通鉴》卷257）&lt;/p&gt;
#   887年: 周迪妻某氏。迪善贾，往来广陵。会毕师铎乱，人相掠卖以食。迪饥将绝，妻曰：“今欲归，不两全。君亲在，不可并死，愿见卖以济君行。”迪不忍，妻固与诣肆，售得数千钱以奉。迪至城门，守者谁何，疑其绐，与迪至肆问状，见妻首已在枅矣。迪里余体归葬之。（《新唐书·卷二百五·列传第一百三十·列女》㉕*）
#   888年: （李）罕之与（张）言甚笃，然性猜暴。是时大乱后，野无遗秆，部卒日剽人以食。《新唐书·卷一百八十七·列传第一百一十二·二王诸葛李孟》㉕*）&lt;p&gt;时大乱之后，野无耕稼，罕之部下以俘剽为资，啖人作食。……自是罕之日以兵寇钞怀、孟、晋、绛，数百里内，郡邑无长吏，闾里无居民。……自是数州之民，屠啖殆尽，荆棘蔽野，烟火断绝，凡十余年。（《旧五代史·卷十五（梁书）·列传五》㉕）&lt;/p&gt;&lt;p&gt;罕之留其子颀事晋，乃之泽州，日以兵钞怀、孟间，啖人为食。（《新五代史·卷四十二·杂传第三十·朱宣等》㉕）&lt;/p&gt;&lt;p&gt;（李）罕之勇而无谋，性复贪暴，意轻（张）全义，闻其勤俭力穑，笑曰：“此田舍一夫耳！”…….（李）罕之所部不耕稼，专以剽掠为资，啖人为粮。……（李罕之）以寇钞为事，自怀、孟、晋、绛数百里间，州无刺史，县无令长，田无麦禾，邑无烟火者，殆将十年。（《资治通鉴》）&lt;/p&gt;
#   889年，[[:w:唐昭宗|唐昭宗]]龍紀元年：楊行密圍宣州，城中食盡，人相啖……（《資治通鑒》卷258）
#   891年: 会吏盗减诸军禀食，（王）建怒其众曰：“招讨吏之谋也。”纵士执之，醢食于军。（《新唐书·卷二百二十四下·列传第一百四十九下·叛臣下》㉕*）&lt;p&gt;一日，（王）建阴令军士于行府门外擒（韦）昭度亲吏，脔而食之，（王）建徐启（韦）昭度曰：“盖军士乏食，以至于是耶！”昭度大惧，遂留符节与建，即日东还。（《旧五代史·卷一百三十六·僭伪列传三》㉕）&lt;/p&gt;&lt;p&gt;昭度迟疑未决，建遣军士擒昭度亲吏于军门，脔而食之，建入白曰：“军士饥，须此为食尔！”昭度大恐，即留符节与建而东。（《新五代史·卷六十三·前蜀世家第三》㉕）&lt;/p&gt;&lt;p&gt;庚子，（王）建阴令东川将唐友通等擒（韦）昭度亲吏骆保于行府门，脔食之，云其盗军粮。（《资治通鉴》卷258）&lt;/p&gt;
#   891年: 孙儒悉焚扬州庐舍，尽驱丁壮及妇女渡江，杀老弱以充食。（《资治通鉴》卷258）
#   893年: 景福二年春，（李克用）大举以伐王镕，……王镕出师三万来援，武皇（李克用）逆战于叱日岭下，镇人败，斩首万余级。时岁饥，军乏食，脯尸肉而食之。（《旧五代史·卷二十六（唐书）·武皇纪下》㉕*）&lt;p&gt;（李克用的）河东军无食。脯其尸而啖之。 （《资治通鉴》卷259）&lt;/p&gt;
#   894年: 王建攻彭州，城中人相食（《资治通鉴》卷259）
#   902年，唐昭宗天复二年：是冬，大雪，（凤翔）城中食尽，冻馁死者不可胜计，或卧未死，肉已为人所。市中卖人肉斤直钱百，犬肉值五百。”（《资治通鉴》卷263）&lt;p&gt;昭宗在凤翔，为梁兵所围，城中人相食，父食其子，而天子食粥，六宫及宗室多饿死。其穷至于如此，遂以亡。（《新唐书·卷五十二·志第四十二·食货二》㉕*）&lt;/p&gt;&lt;p&gt;（朱温的后）梁军围之（凤翔）逾年，（李）茂贞每战辄败，闭壁不敢出。城中薪食俱尽，自冬涉春，雨雪不止，民冻饿死者日以千数。米斗直钱七千，至烧人屎煮尸而食。父自食其子，人有争其肉者，曰：“此吾子也，汝安得而食之！”人肉斤直钱百，狗肉斤直钱五百。父甘食其子，而人肉贱于狗。天子于宫中设小磨，遣宫人自屑豆麦以供御，自后宫、诸王十六宅，冻馁而死者日三四。城中人相与邀遮茂贞，求路以为生。（《新五代史·卷四十·杂传第二十八·李茂贞等》㉕）&lt;/p&gt;

==五代十國==
#   906年:天祐三年，（朱）全忠自将攻沧州，……全忠环沧筑而沟之，内外援绝，人相食。（刘）仁恭求战，不许。（《新唐书· 卷二百一十二·列传第一百三十七·藩镇卢龙》㉕*）&lt;p&gt;汴人深沟高垒以攻沧州，内外阻绝，（刘）仁恭不能合战，城中大饥，人相篡啖，析骸而爨，丸土而食，转死骨立者十之六七。……城中乏食，米斗直三万，人首一级亦直十千，军士食人，百姓食墐土，驴马相遇，食其鬃尾，士人出入，多为强者屠杀。（《旧五代史·卷一百三十五·僭伪列传二》㉕）&lt;/p&gt;&lt;p&gt;梁军壁长芦，深沟高垒，（刘）仁恭不能近。沧州被围百余日，城中食尽，人自相食，析骸而爨，或丸墐土而食，死者十六七。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕）&lt;/p&gt;&lt;p&gt;时汴军筑垒围沧州，鸟鼠不能通。（刘）仁恭畏其强，不敢战。城中食尽，丸土而食，或互相掠啖。（《资治通鉴》卷265）&lt;/p&gt;
#   909年:（刘）守文将吏孙鹤、吕兖等，立守文子延祚以距（刘）守光，守光围之百余日，城中食尽，米斛直钱三万，人相杀而食，或食墐土，马相食其骏尾，（吕）兖等率城中饥民食以麹，号“宰务”，日杀以饷军。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕*）&lt;p&gt;刘守光围沧州久不下，执刘守文至城下示之，犹固守。城中食尽，民食堇泥，军士食人，驴马相啖尾。吕兖选男女羸弱者，饲以黮面而烹之，以给军食，谓之宰杀务。 （《资治通鉴》卷267）&lt;/p&gt;
#  911: （刘）守光大怒，推之（孙鹤）伏锧，令军士割其肉生啖之。鹤大呼曰：“百日之外，必有急兵矣！”守光命窒其口，寸斩之，有识为之嗟惋。（《旧五代史·卷一百三十五·僭伪列传二》㉕*）&lt;p&gt;（刘）守光怒，推之（孙鹤）伏锧，令军士割而啖之。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕）&lt;/p&gt;&lt;p&gt;（刘）守光怒，伏诸质上，令军士剐而啖之。鹤呼曰：“百日之外，必有急兵！”守光命以土窒其口，寸斩之。（《资治通鉴》卷268）&lt;/p&gt;
#  916: 晋人围贝州逾年，城中食尽，啖人为粮。（《资治通鉴》卷269）
#   922年: （李存勖）获（张）处球、处瑾、处琪并其母，及同恶高濛李翥、齐俭等，皆折足送行台，镇人请醢而食之；（《旧五代史·卷二十九（唐书）·庄宗纪三》㉕*）
#  925年，後唐莊宗同光三年： （郭）崇韬欲诛（王）宗弼以自明，己巳，白（李）继岌收宗弼及王宗勋、王宗渥，皆数其不忠之罪，族诛之，籍没其家。蜀人争食宗弼之肉。 （《资治通鉴》卷274）
#   929年: （董璋）遣其将李彦钊扼剑门关为七砦，于关北增置关，号永定。凡唐戍兵东归者，皆遮留之，获其逃者，覆以铁笼，火炙之，或刲肉钉面，割心而啖。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕*）
#  930: （董）璋怒，令军士十人，持刀刲割其（姚洪）肤，燃镬于前，自取啖食，洪至死大骂不已。（《旧五代史·卷七十（唐书）·列传二十二》㉕*）&lt;p&gt;（董）璋怒，然镬于前，令壮士十人刲其肉而食，洪至死大骂。（《新五代史·卷三十三·死事传第二十》㉕）&lt;/p&gt;&lt;p&gt;（董）璋怒，然镬于前，令壮士十人刲其（姚洪）肉自啖之，洪至死骂不绝声。（《资治通鉴》卷277）&lt;/p&gt;
#   约930年:（李）赞华好饮人血，姬妾多刺臂以吮之；婢仆小过，或抉目，或刀刲火灼；夏氏不忍其残，奏离婚为尼。 （《资治通鉴》卷277）
#  934: （薛）文杰善数术，自占云：“过三日可无患。”送者闻之，疾驰二日而至，军士踊跃，磔文杰于市，闽人争以瓦石投之，脔食立尽。（《新五代史·卷六十八·闽世家第八》㉕*）&lt;p&gt;（薛）文杰出，（王）继鹏伺之于启圣门外，以笏击之仆地，槛车送军前，市人争持瓦砾击之。文杰善术数，自云过三日则无患。部送者闻之，倍道兼行，二日而至，士卒见之踊跃，脔食之（《资治通鉴》卷278）&lt;/p&gt;
#   约942年: （石）信所至黩货，好行杀戮。军士有犯法者，信召其妻子，对之刲剔支解，使自食其肉，血流盈前，信命乐饮酒自如也。（《新五代史·卷十八·汉家人传第六》㉕*）
#   944年: 同（州）、华（州）奏，人民相食。（《旧五代史·卷七十（唐书）·列传二十二》㉕*）
#   944年: （后晋少帝石重贵）命李守贞、符彦卿率师东讨。（杨）光远素无兵众，惟婴城（青州）自守，守贞以长连城围之。冬十一月，（杨）承勋与弟承信、承祚见城中人民相食将尽，知事不济，劝（杨）光远乞降，冀免于赤族。（《旧五代史·卷九十七（晋书）·列传十二》㉕*）&lt;p&gt;契丹已北，出帝（石重贵）复遣（李守贞、符彦卿东讨，光远婴城固守，自夏至冬，城中人相食几尽。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕）&lt;/p&gt;
#  945年: 闽人或告福州援兵谋叛，闽主（王）延政收其铠仗，遣还，伏兵于隘，尽杀之，死者八千馀人，脯其肉以归为食。 （《资治通鉴》卷284）
#  947年: （杨）承勋事晋为郑州防御使，（耶律）德光灭晋，使人召承勋至京师，责其劫父，脔而食之。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕*）&lt;p&gt;戊子，（辽军）执郑州防御使杨承勋至大梁，责以杀父叛契丹，命左右脔食之。（《资治通鉴》卷286）&lt;/p&gt;
#  947年，后晋天福十二年（947年：大同元年春正月……己丑，以张彦泽擅徙重贵开封，杀桑维翰，纵兵大掠，不道，斩于市。晋人脔食之。（《辽史· 卷四·本纪第四·太宗下》㉕*）&lt;p&gt;戎王（辽太宗耶律德光）知其（张彦泽）众怒，遂令弃市，仍令高勋监决，断腕出锁，然后刑之。勋使人剖其心以祭死者，市人争其肉而食之。（《旧五代史·卷九十八（晋书）·列传十三》㉕）&lt;/p&gt;&lt;p&gt;百官皆请不赦（张彦泽），而都人争投状疏其恶，乃命高勋监杀之。彦泽前所杀士大夫子孙，皆缞绖杖哭，随而诟詈，以杖朴之，彦泽俯首无一言。行至北市，断腕出锁，然后用刑，勋剖其心祭死者，市人争破其脑，取其髓，脔其肉而食之。（《新五代史·卷五十二·杂传第四十·杜重威等》㉕）&lt;/p&gt;&lt;p&gt;己丑，斩（张）彦泽、（傅）住皃于北市，仍命高勋监刑。彦泽前所杀士大夫子孙，皆绖杖号哭，随而诟詈，以杖扑之。勋命断腕出锁，剖其心以祭死者。市人争破其脑取髓，脔其肉而食之。 （《资治通鉴》卷286）&lt;/p&gt;
#  948年: （苏）逢吉等秘不发丧，下诏称：“（杜）重威父子，因朕小疾，谤议摇众，皆斩之。”磔死于市，市人争啖其肉。（《旧五代史·卷一百（汉书）·高祖纪下》㉕*）&lt;p&gt;磔（杜）重威尸于市，市人争啖其肉，吏不能禁，斯须而尽。 （《资治通鉴》卷287）&lt;/p&gt;
#   948年: （李）守贞自谓天时人事合符于己，乃潜结草贼，令所在窃发，遣兵据潼关。朝廷命白文珂、常思等领兵问罪，复遣枢密使郭威西征。……既而城中粮尽，杀人为食。（《旧五代史·卷一百九（汉书）·列传六》㉕*）&lt;p&gt;（李）守贞（潼关）城中兵无几，而食又尽，杀人而食。（《新五代史·卷五十二·杂传第四十·杜重威等》㉕）&lt;/p&gt;
#  949年，後漢高祖乾佑元年二年：（赵）思绾粮尽，城中人相食（宋）（《宋史· 卷二百五十二·列传第十一·王景等》㉕*）&lt;p&gt;朝廷闻之，命郭从义、王峻帅师伐之（赵思绾）。及攻其城（长安），王师伤者甚众，乃以长堑围之。经年粮尽，遂杀人充食。思绾尝对众取人胆以酒吞之，告众曰：“吞此至一千，即胆气无敌矣。”（《太平广记》：贼臣赵思绾自倡乱至败，凡食人肝六十六，无不面剖而脍之。）（《旧五代史·卷一百九（汉书）·列传六》㉕）&lt;/p&gt;&lt;p&gt;隐帝（后汉隐帝刘承祐）遣郭威西督诸将兵，先围（李）守贞于河中。居数月，（赵）思绾城中食尽，杀人而食，每犒宴，杀人数百，庖宰一如羊豕。思绾取其胆以酒吞之，语其下曰：“食胆至千，则勇无敌矣！” （《新五代史·卷五十三·杂传第四十一·王景崇等》㉕）&lt;/p&gt;&lt;p&gt;赵思绾好食人肝，常面剖而脍之，脍尽，人犹未死。又好以酒吞人胆，谓人曰：吞此千数，则胆无敌矣。长安城中食尽，取妇女幼稚为军粮，日计数而给之。每犒军，辄屠数百人，如羊豖法。（《资治通鉴》卷288）&lt;/p&gt;
#  950年: （马希萼）脔食李弘皋、（李）弘节、唐昭胤、杨涤。（《资治通鉴》）
#   苌从简（后唐、后晋武将），陈州人也。……好食人肉，所至多潜捕民间小儿以食。（《新五代史·卷四十七·杂传第三十五·华温琪等》㉕*）
#  [[:w:吴国 (五代十国)|吳國]]將領[[:w:高澧|高澧]]「嗜殺人而飲血，日暮，必於宅前，後掠行人而食之」。（《南村辍耕录》引《九国志》）

==辽宋金==
从《宋史》开始，二十五史开始频繁记载割肉疗亲的尽孝的故事，这反映了儒家伦理和人肉治病理念的普及，宋朝官方是褒奖这种做法的，之后元朝法律禁止，明清官方态度有所保留，但屡禁不止，愈演愈烈。

* 冠冕百行莫大于孝，范防百为莫大于义。先王兴孝以教民厚，民用不薄；兴义以教民睦，民用不争。率天下而由孝义，非履信思顺之世乎。太祖、太宗以来，子有复父仇而杀人者，壮而释之；刲股割肝，咸见褒赏；至于数世同居，辄复其家。一百余年，孝义所感，醴泉、甘露、芝草、异木之瑞，史不绝书，宋之教化有足观者矣。作《孝义传》。《宋史· 卷四百五十六·列传第二百一十五·孝义》

岳飞《满江红》的“壮志饥餐胡虏肉，笑谈渴饮匈奴血”可能是大众文化中最广泛流传的称赞吃人的文学作品。

#  辽穆宗时期（951年-969年）：初，女巫肖古上延年药方，当用男子胆和之。不数年，杀人甚多，至是（957年，应历七年），觉其妄，辛巳，射杀之。（《辽史·卷六·本纪第六·穆宗上》㉕*）&lt;p&gt;京师置百尺牢以处系囚。盖其（辽穆宗）即位未久，惑女巫肖古之言，取人胆合延年药，故杀人颇众。后悟其诈，以鸣镝丛射、骑践杀之。（《辽史·卷六十一·志第三十·刑法志上》㉕）&lt;/p&gt;
#   963年: 众皆感愤，遂破其众于平津亭，擒（张）文表脔而食之。（《宋史· 卷四百八十三·列传第二百四十二·世家六》㉕*）
#  963年乾德元年：（李）处耘释所俘体肥者数十人，令左右分啖之，黥其少健者，令先入朗州。 （《宋史· 卷二百五十七·列传第十六· 吴廷祚等》㉕*）
#  969年，開寶二年（969）：[[:w:王彥昇|王彥昇]]改防州防御使，是冬，又移原州（甘肅鎮原）。 西人（甘肅少數民族）有犯漢法者，彥升不加刑，召僚屬飲宴，引所犯，以手捽斷其耳，大嚼，巵酒下之。其人流血被體，股栗不敢動。前後啗者數百人。西人畏之，不敢犯塞。（[[:w:王辟之|王辟之]]《澠水燕談錄》，《宋史·卷二百五十·列传第九·王彥昇》㉕*）
#  970年，开宝三年：命分司西京。（王）继勋残暴愈甚，强市民家子女备给使，小不如意，即杀食之，而棺其骨弃野外。……长寿寺僧广惠常与继勋同食人肉，令折其胫而斩之。洛民称快。（《宋史· 卷四百六十三·列传第二百二十二·外戚上》㉕*）
#  1006年: 三年，（德恭）被疾，子承庆刲股肉食之。（《宋史· 卷二百四十四·列传第三·宗室一》㉕*）
#  1048年，[[:w:宋仁宗|宋仁宗]]庆历八年：明年，河北大饥，人相食，（子）鼎经营赈救，颇尽力。（《宋史·卷三百·列传第五十九·杨偕等》㉕*）&lt;p&gt;河北、京東西大水為災，人相食，流民入京東者不可勝數（《[[:w:續資治通鑑|續資治通鑑]]》卷50）&lt;/p&gt;
#  约1053年，宋仁宗时期：[[:w:侬智高|（侬）智高]]母[[:w:阿侬|阿侬]]有计谋，智高攻陷城邑，多用其策，僭号皇太后，性惨毒，嗜小儿肉，每食必杀小儿。（《宋史· 卷四百九十五·列传第二百五十四·蛮夷三》㉕*） 
#  1087年，[[:w:宋哲宗|宋哲宗]]元祐二年，[[:w:苏辙|苏辙]]《因旱乞许群臣面对言事剳子》：“臣伏见二年以来，民气未和，天意未顺，災沴荐至，非水即旱。淮南饥饉，人至相食。河北流移，道路不绝。京东困弊，盗贼群起。二圣遇災忧惧，顷发仓廪以救其乏绝，独此三路所散，已仅三百万斛矣！異时赈賉未见此比。然而民力已困，国用己竭，而旱势未止，夏麦失望，秋稼未立，数月之后，公私无继，群盗蜂起，势有必至，臣未知朝廷何以待此？……”
#  1102年: （高永年）行三十里，逢羌帐下亲兵，皆永年昔所推纳熟户也。永年不之备，羌遽执永年以叛，遂为多罗巴所杀，探其心肝食之，谓其下曰：“此人夺我国，使吾宗族漂落无处所，不可不杀也。”（《宋史· 卷四百五十三·列传第二百一十二·忠义八》㉕*）
#  1118年，辽天庆八年（宋重和元年，1118年），十二月，“宁昌军（治懿州）节度使刘宏（无可考）以懿州（治宁昌，今阜新市东北之塔营子村）户三千降金。时山前诸路（此指辽东，非燕山之南）大饥，乾（辽宁北镇南）显（北镇北）宜（义县）锦（锦州市）兴中（朝阳市）等路，斗粟值数缣，民削榆皮食之，既而人相食。”（《辽史· 卷二十八·本纪第二十八·天祚皇帝二》㉕*）
#  1121年: 贼（霍成富）怒，脔其（詹良臣）肉，使自啖之。良臣吐且骂，至死不绝声，见者掩面流涕，时年七十二。（《宋史· 卷四百四十六·列传第二百五·忠义一》㉕*）
#  “甲辰宣和六年（1124年）时转粮给燕山（府治北京西南）民力疲困，重以盐额科敛，加之连年凶荒，民食榆皮野菜不给，至自相食。于是饥民并起为盗。山东有张万仙者，众十万，号敢炽。张迪者，众五万，围濬州（濬州，平川军，治滑州黎阳）五日而去。濬州去京纔一百六十里，而初不知。河北有高托山者，号三十万。其余一二万者，不可胜计也。”（《九朝编年备要卷二十九》）
#  [[:w:宋徽宗|宋徽宗]]宣和七年（1125年）十二月，金两路攻宋。王禀皆破之，“然人众乏粮，三军先食牛马骡，次烹弓弩皮甲，百姓煮萍实、糠籺、草茭以充腹，既而人相食。[九月]城破，禀犹率羸卒巷战，突围出，金兵追之急，遂负太原庙中太宗御容赴汾水死，子荀殉之。”（《续资治通鉴卷九十七》）
#  1125年: 刘敏行，平州人。登天会三年进士。除太子校书郎，累迁肥乡令。岁大饥，盗贼掠人为食。诸县老弱入保郡城，不敢耕种，农事废，畎亩荒芜。（《金史· 卷一百二十八·列传第六十六·循吏》㉕*）
#  1129年：（建炎）三年，山东郡国大饥，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1131年: 有孙知微者，以朝请大夫通判舒州。绍兴元年，贼刘忠入其境，执知微以去，知微不屈，忠怒，脔而食之。（《宋史· 卷四百五十三·列传第二百一十二·忠义八》㉕*）
#  1131年：五湖捕鱼人夏宁聚众千余，掠人为食，郭仲威余党出没淮南，邵青据通州，光世皆招降之。（《宋史·卷三百六十九·列传第一百二十八·张俊》㉕*）&lt;p&gt;五湖捕魚人夏寧，“聚其徒為盜，後有眾千餘，專掠人以為食，……寧等無食，半月之間復啖萬餘人，是日，始具舟迎之。由是江北鄉村愈覺凋殘矣。”（《续资治通鉴卷一零九》）&lt;/p&gt;
#  约1133年，宋高宗紹興三年：唐初，贼朱粲以人为粮，置捣磨寨，谓“啖醉人如食糟豚”。每览前史，为之伤叹。而自靖康丙午岁，金人乱华，六七年间，山东、京西、淮南等路，荆榛千里，斗米至数十千，且不可得。盗贼、官兵以至居民，更互相食。人肉之价，贱于犬豕，肥壮者一枚不过十五千，全躯暴以为腊。登州范温率忠义之人，绍兴癸丑岁泛海到钱唐，有持至行在犹食者。老瘦男子 词谓之“饶把火”，妇人少艾者名为“不羡羊”，小儿呼为“和骨烂”，又通目为“两脚羊”。唐止朱粲一军，今百倍于前世，杀戮焚溺饥饿疾疫陪堕，其死已众，又加之以相食。杜少陵谓“丧乱死多门”，信矣！不意老眼亲见此时，呜呼痛哉！ （莊綽《雞肋編》）
#  [[:w:宋宁宗|宋宁宗]]嘉定年間，[[:w:林千之|林千之]]任西欽州知州，得了一种病（末疾），有個醫士告訴他，吃童女的肉可以強筋健骨。于是，林千之派人在本州境內捕少女，制成肉乾，叫做“地雞”。&lt;ref&gt;王永寬《中國古代酷刑》&lt;/ref&gt;
#  1210年:（嘉定）三年春，建康府大飢，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1215年: 此數人者（李全等造反者），出沒島崓，寶貨山委而不得食，相率食人。（《宋史· 卷四百七十六·列傳第二百三十五·叛臣中》㉕*）
#  1215年: 乙亥，中都降。（王）檝进言曰：“国家以仁义取天下，不可失信于民，宜禁虏掠，以慰民望。”时城中绝粒，人相食，乃许军士给粮，入城转粜，故士得金帛，而民获粒食。（《元史· 卷一百五十三·列传第四十·刘敏等》㉕*）
#  1216: 是春，河朔人相食。（《金史· 卷二十三·志第四·五行》㉕*）&lt;p&gt;四年，河北行省侯摯言：“河北人相食，觀、滄等州鬥米銀十餘兩。（《金史· 卷五十·志第三十一·食貨五》㉕）&lt;/p&gt;&lt;p&gt;金人迁汴，河朔盗起，……太师、国王木华黎兵至城下，……是时兵乱，民废农耕，所在人相食。（《元史· 卷一百五十一·列传第三十八·薛塔剌海等》㉕）&lt;/p&gt;
#  1216年: 邸顺，保定行唐人，岁甲戌，（邸顺）率众来归（元），（元）太祖授行唐令。……丙子，真定饥，群盗据城叛，民皆穴地以避之，盗发地而啖其人，顺擒数百人杀之。（《元史· 卷一百五十一·列传第三十八·薛塔剌海等》㉕*）
#  1224: 十一月……壬子，京城人相食。癸醜，詔曹門、宋門放士民出就食。（《金史· 卷十八·本紀第十八·哀宗下》㉕*）
#  1227年: 时（李）全在围一年，食牛马及人且尽，将自食其军。初军民数十万，至是余数千矣。（《宋史· 卷四百七十七·列传第二百三十六·叛臣下》㉕*）
#  1228年: （完颜）白撒辈纵军四出，剽掠俘虏，挑掘焚炙，靡所不至。哭声相接，尸骸盈野。都尉高禄谦、苗用秀辈仍掠人食之，而白撒诛斩在口，所过官吏残虐不胜，一饭之费有数十金不能给者，公私皇皇，日皆徯大兵至矣。（《金史· 卷一百十三·列传第五十一·完颜赛不等》㉕*）
#  1232年: 时汴京内外不通，米升银二两。百姓粮尽，殍者相望，缙绅士女多行乞于市，至有自食其妻子者，至于诸皮器物皆煮食之，贵家第宅、市楼肆馆皆撤以爨。（《金史· 卷一百十五·列传第五十三·完颜奴申等》㉕*）
#  1233年，绍定六年（1233年）：（南宋大将[[:w:史嵩之|史嵩之]]围唐州，）城中粮尽，人相食，金将乌库哩黑汉，杀其爱妾以啖士，士争杀其妻子（《金史· 卷一百二十三·列传第六十一·忠义三》㉕*，《续资治通鉴·宋纪》）&lt;p&gt;乙酉，大元召宋兵攻唐州，元帅右监军乌古论黑汉死于战，主帅蒲察某为部曲兵所食。城破，宋人求食人者尽戮之，余无所犯。（《金史· 卷十八·本纪第十八·哀宗下》㉕）&lt;/p&gt;
#  1233: 国用安，先名安用，本名咬儿，淄州人。红袄贼杨安儿、李全余党也。……移兵攻徐，（国）用安投水死，求得其尸，剖面系马尾，为怨家田福一军脔食而尽。（《金史· 卷一百十七·列传第五十五·徒单益都等》㉕*）
#  1234年: 端平元年正月辛丑，黑气压（蔡州）城上，日无光，降者言：“城中绝粮已三月，鞍靴败鼓皆糜煮，且听以老弱互食，诸军日以人畜骨和芹泥食之，又往往斩败军全队，拘其肉以食，故欲降者众。”（《宋史· 卷四百一十二·列传第一百七十一·孟珙》㉕*）
#  1234年:甲午，蔡州破，金主自焚死。时汴梁受兵日久，岁饥，人相食，速不台下令纵其民北渡以就食。（《元史· 卷一百二十一·列传第八·速不台》㉕*）
#  约1237: 岁大饥，人相食，留守别之杰讳不诘，（徐）鹿卿命掩捕食人者，尸诸市。（《宋史· 卷四百二十四·列传第一百八十三·陆持之》㉕*）
#  1272年:咸淳七年，江南大饥。八年冬，襄阳饥，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1276: 德祐二年正月，扬州饥。三月，扬州谷价腾踊，民相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）&lt;p&gt;阿术攻扬（州）久不拔，乃筑长围困之。冬，城中食尽，死者满道。明年二月，饥益甚，赴濠水死者日数百，道有死者，众争割啖之立尽。……兵有烹子而食者，犹日出苦战。（《宋史·卷四百二十一·列传第一百八十·杨栋等》㉕）&lt;/p&gt;
#  1277: 十一月，泸州食尽，人相食，遂破之，安抚王世昌自经死。（《宋史· 卷四百五十一·列传第二百一十·忠义六》㉕*）
#  益州双流人周善敏，丧父，庐于墓侧。母病，又割股肉以啖之，遂愈。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  杨庆，鄞人。父病，贫不能召医，乃刲股肉啖之，良已。其后母病不能食，庆取右乳焚之，以灰和药进焉，入口遂差，久之乳复生。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  （伊）审征幼以孝闻，母病，割股肉啖之。（《宋史· 卷四百七十九·列传第二百三十八·世家二》㉕*）
#  刘孝忠，并州太原人。母病经三年，孝忠割股肉、断左乳以食母；（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  吕升，莱州人。父权失明，剖腹探肝以救父疾，父复能视而升不死。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  成象，渠州流江人。以诗书训授里中，事父母以孝闻。母病，割股肉食之，诏赐束帛醪酒。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  庞天祐，江陵人。以经籍教授里中。父疾，天祐割股肉食之；（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  张伯威，大安军人。……大母黄，年九十八，不忍之官。黄得血痢疾濒殆，伯威剔左臂肉食之，遂愈。继母杨因姑病笃，惊而成疾，伯威复剔臂肉作粥以进，其疾亦愈。伯威妹嫁崔均，其姑王疾，妹亦剔左臂肉作粥以进，达旦即愈。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  母病，（奎）辄割股肉和药以进，母遂愈。（《宋史· 卷三百二十四·列传第八十三·石普》㉕*）
#  （张）掞幼笃孝，蕴病，刲股肉以疗。（《宋史· 卷三百三十三·列传第九十二·杨佐等》㉕*）
#  （常）真妻病，子晏割股肉以养母（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  有朱云孙妻刘氏，姑病，云孙刲股肉作糜以进而愈。姑复病，刘亦刲股以进，又愈。尚书谢谔为赋《孝妇诗》。（《宋史· 卷四百六十·列传第二百一十九·列女》㉕*）
#  聂孝女，字舜英，尚书左右司员外郎天骥之长女也。……崔立劫杀宰相，天骥被创甚，日夜悲泣，恨不即死。舜英谒医救疗百方，至刲其股杂他肉以进，而天骥竟死。时京城围久食尽，……葬其父之明日，绝脰而死。一时士女贤之，有为泣下者。（《金史· 卷一百三十·列传第六十八·列女》㉕*）
#   呼延赞，并州太原人。……其子尝病，赞刲股为羹疗之。（《宋史·卷二百七十九·列传第三十八· 王继忠等》㉕*）
#   蒋偕，字齐贤，华州郑县人。幼贫，有立志。父病，尝刲股以疗，父愈，诘之曰：“此岂孝邪？”曰：“情之所感，实不自知也。”（《宋史·卷三百二十六·列传第八十五·景泰等》㉕*）
#   邑人朱氏女刲股愈母疾，人颂传之，以为治化所致。（《宋史·卷三百四十八·列传第一百七·傅楫等》㉕*）
#   甲幼孤多难，母病，刲股以进。（《宋史·卷三百九十七·列传第一百五十六·徐谊等》㉕*）
#   赵葵，字南仲，京湖制置使方之子。……葵母疾，谒告省侍不得，刲股杂药以寄之。母卒，葵求解官，不许，不得已，卒哭复视事。（《宋史·卷四百一十七·列传第一百七十六·乔行简等》㉕*）
#   陈宗，永嘉人。年十六，母蔡病笃，刲股为饵，病愈。已而复病不救，宗一恸而绝。（《宋史·卷四百五十六·列传第二百一十五·孝义》㉕*）
#   吕仲洙女，名良子，泉州晋江人。父得疾濒殆，女焚香祝天，请以身代，刲股为粥以进。（《宋史·卷四百六十·列传第二百一十九·列女》㉕*）

==元==
元朝法律禁止割肉疗亲，“诸为子行孝，辄以割肝、刲股、埋儿之属为孝者，并禁止之。（《元史· 卷一百五·志第五十三·刑法四》）”但《元史》记载了诸多此般事迹，可见屡禁不止，可能也反映了蒙汉的文化差异。

#   1262年:（中统三年），五月庚申，筑环城（济南）围之；甲戌，围合。（李）鋋自是不得复出，……分军就食民家，发其盖藏以继，不足，则家赋之盐，令以人为食。（《元史·卷二百六·列传第九十三·叛臣》㉕*）
#   1301: 行省右丞刘深远征八百媳妇国，此乃得已而不已之兵也。彼荒裔小邦，远在云南之西南又数千里，……深欺上罔下，帅兵伐之，经过八番，纵横自恣，恃其威力，虐害居民，中途变生，所在皆叛。深既不能制乱，反为乱众所制，军中乏粮，人自相食，（《元史·卷一百六十八·列传第五十五·陈祐（天祥）等》㉕*）
#   1308年:（至大元年六月）河南、山东大饥，有父食其子者，以两道没入赃钞赈之。（《元史· 卷二十二·本纪第二十二·武宗一》㉕*）
#  1319年:延佑六年秋七月丙辰，“来安路总管岑世兴叛，据唐兴州”，杀兼州知州[[:w:黄克仁|黄克仁]]，分食其尸。&lt;ref&gt;《新元史·卷二百四十八·列传第一百四十五》；《招捕总录》&lt;/ref&gt;
#   约1329年: 贼稍引去，（褚不华）乃出，抵杨村桥，贼奄至，杀廉访副使不达失里，啖其尸。（《元史· 卷一百九十四·列传第八十一·忠义二》㉕*）
#   约1329年: （褚）不华以余兵入淮安。……城中饿者仆道上，即取啖之，一切草木、螺蛤、鱼蛙、燕乌，及靴皮、鞍韂、革箱、败弓之筋皆尽，而后父子夫妇老稚更相食，撤屋为薪，人多露处，坊陌生荆棘。力既尽，城陷。（《元史· 卷一百九十四·列传第八十一·忠义二》㉕*）
#   1328年: （天历元年十二月）陕西自泰定二年至是岁不雨，大饥，民相食。（《元史· 卷三十二·本纪第三十二·文宗一》㉕*）&lt;p&gt;天历元年八月，陕西大旱，人相食。（《元史· 卷五十·志第三上·五行一》㉕）&lt;/p&gt;
#   1329年: 天历二年，关中大旱，饥民相食。（《元史· 卷一百七十五·列传第六十二·张珪等》㉕*）&lt;p&gt;文宗天历二年三月，屯田总管兼管河渠司事郭嘉议言：“……近因奉元亢旱，五载失稔，人皆相食，流移疫死者十七八。”（《元史· 卷六十五·志第十七上·河渠二》㉕）&lt;/p&gt;&lt;p&gt;天历二年，（乃蛮台）迁陕西行省平章政事。关中大饥，……京兆民掠人而食之，则命分健卒为队，捕强食人者，其患乃已。（《元史· 卷一百三十九·列传第二十六·乃蛮台等》㉕）&lt;/p&gt;
#   1329:（天历二年夏四月）丙辰，行在所遣只儿哈郎等至京师。河南廉访司言：“河南府路以兵、旱民饥，食人肉事觉者五十一人，饿死者千九百五十人，饥者一万七千四百余人。乞弛山林川泽之禁，听民采食，行入粟补官之令，及括江淮僧道余粮以赈。”（《元史· 卷三十三·本纪第三十三·文宗二》㉕*）
#   1338年: 重改至元四年，…. 贼怒，缚景茂于树，脔其肉，使自啖。景茂益愤骂，贼遂以刀决其口，至耳傍，景茂骂不绝声而死。（《元史· 卷一百九十三·列传第八十·忠义一》㉕*）
#   1342年: 二年春正月…..，是月，大同饥，人相食，运京师粮赈之。（《元史· 卷四十·本纪第四十·顺帝三》㉕*）&lt;p&gt;至正二年，彰德、大同二郡及冀宁平晋、榆次、徐沟县，汾州孝义县，忻州皆大旱，自春至秋不雨，人有相食者。（《元史·卷五十一·志第三下·五行二》㉕）&lt;/p&gt;
#   1343年: （至正）三年，卫辉、冀宁、忻州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1344年: （至正四年）六月，河南巩县大雨，伊、洛水溢，漂民居数百家。济宁路兖州，汴梁鄢陵、通许、陈留、临颍等县大水害稼，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1344年: 八月戊午，祭社稷。丁卯，山东霖雨，民饥相食，赈之。（《元史· 卷四十一·本纪第四十一·顺帝四》㉕*）&lt;p&gt;1344年:（至正四年）八月，益都霖雨，饥民有相食者。（《元史·卷五十一·志第三下·五行二》㉕）&lt;/p&gt;
#   1345年: 五年春，东平路须城、东阿、阳谷三县及徐州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1347: 六月，……彰德路大饥，民相食。（《元史· 卷四十一·本纪第四十一·顺帝四》㉕*）
#   1348: 刘秉直，字清臣，大都武清人。至正八年，来为卫辉路总管，……岁大饥，人相食，死者过半，秉直出俸米，倡富民分粟，馁者食之，病者与药，死者与棺以葬。（《元史· 卷一百九十二·列传第七十九·良吏二》㉕*）
#   1349年: （至正）九年春，胶州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#  [[:w:元惠宗|元惠宗]]至正年间，大饑，“淮右军”軍隊開始吃人，“天下兵甲方殷，而淮右之軍嗜食人，以小兒為上，婦女次之，男子又次之。或使坐兩缸間，外逼以火。或於鐵架上生炙。或縛其手足，先用沸湯澆潑，卻以竹帚刷去苦皮。或盛夾袋中，入巨鍋活煮。或卦作事件而淹之。或男子則止斷其雙腿，婦女則特剜其雙乳。酷毒萬狀，不可具言。總名曰「想肉」，以為食之而使人想之也。”&lt;ref&gt;{{Cite web|title=南村輟耕錄 (四部叢刊本)/卷之九 - 維基文庫，自由的圖書館|url=https://zh.wikisource.org/zh-hant/%E5%8D%97%E6%9D%91%E8%BC%9F%E8%80%95%E9%8C%84_(%E5%9B%9B%E9%83%A8%E5%8F%A2%E5%88%8A%E6%9C%AC)/%E5%8D%B7%E4%B9%8B%E4%B9%9D|website=zh.wikisource.org|access-date=2024-05-28|language=zh-Hant}}&lt;/ref&gt;
#   1352年: （至正）十二年，蕲州、黄州大旱，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1353年: 至正十二年，……明年，春夏大饥，人相食，（余阙）乃捐俸为粥以食之，得活者甚众。（《元史· 卷一百四十三·列传第三十·马祖常等》㉕*）
#   1354年: （至正）十四年，怀庆河内县、孟州，汴梁祥符县，福建泉州，湖南永州、宝庆，广西梧州皆大旱。祥符旱魃再见，泉州种不入土，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1354年: 十四年春，浙东台州，江东饶，闽海福州、邵武、汀州，江西龙兴、建昌、吉安、临江，广西静江等郡皆大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1355: 京师大饥，加以疫疠，民有父子相食者。（《元史· 卷四十三·本纪第四十三·顺帝六》㉕*）
#   1358年: 十八年春，莒州蒙阴县大饥，斗米金一斤。冬，京师大饥，人相食，彰德、山东亦如之。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1358年: （至正）十八年春，蓟州旱。莒州、滨州、般阳淄川县、霍州、鄜州、凤翔岐山县春夏皆大旱。莒州家人自相食，岐山人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1358年: 顺德九县民食蝗，广平人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1359年: （至正）十九年，大都霸州、通州，真定，彰德，怀庆，东昌，卫辉，河间之临邑，东平之须城、东阿、阳谷三县，山东益都、临淄二县，潍州、胶州、博兴州，大同、冀宁二郡，文水、榆次、寿阳、徐沟四县，沂、汾二州，及孝义、平遥、介休三县，晋宁潞州及壶关、潞城、襄垣三县，霍州赵城、灵石二县，隰之永和，沁之武乡，辽之榆社、奉元，及汴梁之祥符、原武、鄢陵、扶沟、杞、尉氏、洧川七县，郑之荥阳、汜水，许之长葛、郾城、襄城、临颍，钧之新郑、密县，皆蝗，食禾稼草木俱尽，所至蔽日，碍人马不能行，填坑堑皆盈。饥民捕蝗以为食，或曝干而积之。又罄，则人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   十九年正月至五月，京师大饥，银一锭得米仅八斗，死者无算。通州民刘五杀其子而食之。保定路莩死盈道，军士掠孱弱以为食。济南及益都之高苑，莒之蒙阴，河南之孟津、新安、黾池等县皆大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1359年: 十八年二月，江西陈友谅遣贼党王奉国等，号二十万，寇信州。明年正月，伯颜不花的斤自衢引兵援焉。……时军民唯食草苗茶纸，既尽，括靴底煮食之，又尽，掘鼠罗雀，及杀老弱以食。五月，大破贼兵。（《元史· 卷一百九十五·列传第八十二·忠义三》㉕*）
#   1360: 至正二十年，（丁好礼）遂拜中书参知政事。时京师大饥，天寿节，庙堂欲用故事大宴会，好礼言：“今民父子有相食者，君臣当修省，以弭大患，燕会宜减常度。”不听，乞谢事，乃以集贤大学士致仕，给全俸家居。（《元史· 卷一百九十六·列传第八十三·忠义四》㉕*）
#   1360年: 李仲义妻刘氏，名翠哥，房山人。至正二十年，县大饥，平章刘哈剌不花兵乏食，执仲义欲烹之。仲义弟马儿走报刘氏，刘氏遽往救之，涕泣伏地，告于兵曰：“所执者是吾夫也，乞矜怜之，贷其生，吾家有酱一瓮、米一斗五升，窖于地中，可掘取之，以代吾夫。”兵不从，刘氏曰：“吾夫瘦小，不可食。吾闻妇人肥黑者味美，吾肥且黑，愿就烹以代夫死。”兵遂释其夫而烹刘氏。闻者莫不哀之。（《元史· 卷二百一·列传第八十八·列女二》㉕*）
#   1362年:（至正）二十二年，河南洛阳、孟津、偃师三县大旱，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   萧道寿，京兆兴平人。……母尝有疾，医累岁不能疗，道寿刲股肉啖之而愈。至元八年，赐羊酒，表其门。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   宁猪狗，山丹州人。母年七十余，患风疾，药饵不效，猪狗割股肉进啖，遂愈。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   潭州万户移剌琼子李家奴，九岁，母病，医言不可治，李家奴割股肉，煮糜以进，病乃痊。抚州路总管管如林、浑州民朱天祥，并以母疾刲割股，旌其家。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   孔全，亳州鹿邑人。父成病，刲股肉啖之，愈。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   赵荣，扶风人。母强氏有疾，荣割股肉啖之者三。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   胡伴侣，钧州密县人。其父实尝患心疾数月，几死，更数医俱莫能疗。伴侣乃斋沐焚香，泣告于天，以所佩小刀于右胁傍刲其皮肤，割脂一片，煎药以进，父疾遂瘳，其伤亦旋愈。朝廷旌表其门。（《元史· 卷一百九十八·列传第八十五·孝友二》㉕*）
#   郎氏，湖州安吉人，宋进士朱甲妻也。……家居，养姑甚谨。姑尝病，郎祷天，刲股肉进啖而愈。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   许氏女，安丰人。父疾，割股啖之乃痊。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   秦氏二女，河南宜阳人，逸其名。父尝有危疾，医云不可攻。姊闭户默祷，凿己脑和药进饮，遂愈。父后复病欲绝，妹刲股肉置粥中，父小啜即苏。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   张义妇，济南邹平人，年十八归里人李伍。……张独家居，养舅姑甚至。父母舅姑病，凡四刲股肉救不懈。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   武用妻苏氏，真定人，徙家京师。用疾，苏氏刲股为粥以进，疾即愈。（《元史· 卷二百一·列传第八十八·列女二》㉕*）

==明==
[[:w:李時珍|李時珍]]完成《本草綱目》，他蒐集藥名是為了「凡經人用者，皆不可遺」，「人部」舉凡毛髮、指甲、牙齒、屎尿、唾液、乳汁、眼淚、汗水、人骨、胞衣（[[:w:紫河車|紫河車]]）、體垢、月水、人勢（陰莖）、人膽、結石……皆可入藥。頭髮可治傷寒、肚疼，男性陰毛治蛇咬，人魄（人吊死級的魂魄）可以安神定魄。

明朝没有像元朝一样法律禁止割肉疗亲，但朱元璋和其礼部尚书公开表示不赞同，但此后仍然多次出现，而且得到政府表彰，还有王族如此做，可见此风难止。

* 至（洪武）二十七年九月，山东守臣言：“日照民江伯儿，母疾，割肋肉以疗，不愈。祷岱岳神，母疾瘳，愿杀子以祀。已果瘳，竟杀其三岁儿。”帝大怒曰：“父子天伦至重。《礼》父服长子三年。今小民无知，灭伦害理，亟宜治罪。”遂逮伯儿，仗之百，遣戍海南。因命议旌表例。礼臣（任亨泰）议曰：“人子事亲，居则致其敬，养则致其乐，有疾则医药吁祷，迫切之情，人子所得为也。至卧冰割股，上古未闻。倘父母止有一子，或割肝而丧生，或卧冰而致死，使父母无依，宗祀永绝，反为不孝之大。皆由愚昧之徒，尚诡异，骇愚俗，希旌表，规避里徭。割股不已，至于割肝，割肝不已，至于杀子。违道伤生，莫此为甚。自今父母有疾，疗治罔功，不得已而卧冰割股，亦听其所为，不在旌表例。”制曰：“可。”（《明史·卷一百三十七·列传第二十五·刘三吾等》）

食人事件的记载：

#  [[:w:韩观|韩观]]杀人甚多，御史欲弹劾他。一日，观召御史饮，以人皮为坐褥，耳目口鼻显然，发散垂褥，首披椅后。肴上，设一人首，观以箸取二目食之，曰：“他禽兽目皆不可食，惟人目甚美。”观前席坐，每拿人至，命斩之，不回首视，已而血流满庭。观曰： “此辈与禽兽不异，斩之如杀虎豹耳。”御史战栗失措曰：“公，神人也。”竟不能劾。&lt;ref&gt;《[[s:湧幢小品/09#韓都督應變|湧幢小品 韓都督應變]]》朱国桢&lt;/ref&gt;
#  1385年，洪武十八年：（韩）林儿本起盗贼，无大志，又听命福通，徒拥虚名。诸将在外者率不遵约束，所过焚劫，至啖老弱为粮，且皆福通故等夷，福通亦不能制。（《明史·卷一百二十二·列传第十·郭子兴 韩林儿》㉕*）
#  约1426年，宣德年间：得（朱）有熺掠食生人肝脑诸不法事，于是并免为庶人。（《明史·卷一百十六·列传第四·诸王》㉕*）
#  1454年，景泰五年：景泰五年，广西古丁等洞贼首蓝伽、韦万山等，纠合蛮类，劫掠南宁、上林、武缘诸处。……贼首韦朝威据古田，县官窜会城，遣典史入县抚谕，烹食之。（《明史·卷三百十七·列传第二百五·广西土司》㉕*）
#  1457年，天顺元年：北畿、山東並飢，發塋墓，斫道樹殆盡。父子或相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  约1465年，成化初：成化初，（彭伦）从赵辅，平大藤峡贼。……（彭）伦大会所部目、把缚俘囚，置高竿，集健卒乱射杀之，复割裂肢体，烹啖诸壮士。（《明史·卷一百六十六·列传第五十四·韩观等》㉕*）
#  1484年，成化二十年：是秋，陝西、山西大旱饑，人相食。停歲辦物料，免稅糧，發帑轉粟，開納米事例振之。（《明史·卷十四·本纪第十四·宪宗二》㉕*）&lt;p&gt;又有虎臣者，麟游人。成化中贡入太学。……省亲归，会陕西大饥，……上言：“臣乡比岁灾伤，人相食，由长吏贪残，赋役失均。请敕有司审民户，编三等以定科徭。”从之。（《明史·卷一百六十四·列传第五十二·邹缉等》㉕）&lt;/p&gt;&lt;p&gt;十六年（何乔新）擢右副都御史，巡抚山西。……进左副都御史。……召拜刑部右侍郎。山西大饥，人相食。命往振，活三十余万人，还流冗十四万户。（《明史·卷一百八十三·列传第七十一·何乔新等》㉕）&lt;/p&gt;&lt;p&gt;汪奎，字文灿，婺源人。……（成化）二十一年，星变，偕同官疏陈十事，言：“……山、陕、河、洛饥民多流郧、襄，至骨肉相啖。请大发帑庾振济，消弭他变。”（《明史·卷一百八十·列传第六十八·张宁等》㉕）&lt;/p&gt;
#  1504年，弘治十七年：十七年，淮、扬、庐、凤洊饥，人相食，且发瘗胔（坟墓尸体）以继之。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1518年，正德十三年：佛郎机，近满剌加。正德中，据满剌加地，逐其王。十三年遣使臣加必丹末等贡方物，请封，始知其名。诏给方物之直，遣还。其人久留不去，剽劫行旅，至掠小儿为食。（《明史·卷三百二十五·列传第二百十三·外国六》㉕*）
#  正德五年（1510年）八月，[[:w:刘瑾|刘瑾]]被磔死，凌迟三日，共剐3300余刀。行刑之日，北京鼎沸，百姓相爭以一钱买刘瑾一塊肉，生吞以泄恨。{{Citation needed}}
#  1519年，正德十四年：是岁，淮、扬饥，人相食。（《明史·卷十六·本纪第十六·武宗》㉕*）&lt;p&gt;十四年三月，有诏南巡，（黄）巩上疏曰：……今江、淮大饥，父子兄弟相食。（《明史·卷一百八十九·列传第七十七·李文祥等》㉕）&lt;/p&gt;&lt;p&gt;（吴）一鹏极陈四方灾异，言：“自去年六月迄今二月，其间天鸣者三，地震者三十八，秋冬雷电雨雹十八，暴风、白气、地裂、山崩、产妖各一，民饥相食二。非常之变，倍于往时。愿陛下率先群工，救疾苦，罢营缮，信大臣，纳忠谏，用回天意。”（《明史·卷一百九十一·列传第七十九·毛澄等》㉕）&lt;/p&gt;
#  1524年，嘉靖三年：三年，湖广、河南、大名、临清饥。南畿诸郡大饥，父子相食，道殣相望，臭弥千里。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;（张）汉卿言：“……今东南洊饥，民至骨肉相食，而搜括之令频行，臣等窃以为不可。”（《明史·卷一百九十二·列传第八十·杨慎》㉕）&lt;/p&gt;&lt;p&gt;世宗即位，（韩邦靖）起山西左参议，分守大同。岁饥，人相食，奏请发帑，不许。（《明史·卷二百一·列传第八十九·陶琰等》㉕）&lt;/p&gt;&lt;p&gt;嘉靖四年二月（余珊）应诏陈十渐，其略曰：……近年以来，黄纸蠲放，白纸催征；额外之敛，下及鸡豚；织造之需，自为商贾。江、淮母子相食，兖、豫盗贼横行，川、陕、湖、贵疲于供饷。（《明史·卷二百八·列传第九十六·张芹等》㉕）&lt;/p&gt;&lt;p&gt;嘉靖初，（湛若水）入朝，……明年进侍读，复疏言：“一二年间，天变地震，山崩川涌，人饥相食，殆无虚月。”（《明史·卷二百八十三·列传第一百七十一·儒林二》㉕）&lt;/p&gt;
#  1529年，嘉靖八年：（杨爵）登嘉靖八年进士，授行人。帝方崇饰礼文，（杨）爵因使王府还，上言：“臣奉使湖广，睹民多菜色，挈筐操刃，割道殍食之。（《明史·卷二百九·列传第九十七·杨最等》㉕*）
#  1549年，嘉靖二十八年：有吴国佐者，洪州司特峒寨苗也，….. 其党石纂太称“太保”，合攻上黄堡，诱败参将黄冲霄，追至永从县，杀守备张世忠，炙而啖之。（《明史·卷二百四十七·列传第一百三十五·刘綎等》㉕*）
#  1552年，嘉靖三十一年：宣、大二镇大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1553年，嘉靖三十二年：京师大饥，人相食，米石二两二钱。（《历代社会风俗事物考》引《金垒子》）
#   1557年，嘉靖三十六年：三十六年，辽东大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1559年，嘉靖三十八年八月：以辽东连年饥馑，至有父食死子者，发银糴粟赈之。（《中外历史年表》）
#   1588，万历十六年：十六年，河南饥，民相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#   1591年，万历十九年：（万历）十九年，（子俊民）还理部事。河南大饥，人相食，请发银米各数十万。（《明史·卷二百十四·列传第一百二·杨博等》㉕*）
#   1593年，万历二十二年：二十二年，河南大饥，人相食，命（钟）化民兼河南道御史往振。荒政具举，民大悦。（《明史·卷二百二十七·列传第一百十五·庞尚鹏等》㉕*）&lt;/p&gt;&lt;p&gt;（陈登云）出按河南。岁大饥，人相食。（《明史·卷二百三十三·列传第一百二十一·姜应麟等》㉕）&lt;/p&gt;
#   1601年，万历二十九年：二十九年，两畿饥。阜平县饥，有食其稚子者。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1611年，万历三十九年：马孟祯，字泰符，桐城人。万历二十六年进士。……三十九年夏，怡神殿灾。孟祯言：“二十年来，郊庙、朝讲、召对、面议俱废，通下情者惟章奏。……畿辅、山东、山西、河南，比岁旱饥。民间卖女鬻儿，食妻啖子，铤而走险，急何能择。”（《明史·卷二百三十·列传第一百十八·蔡时鼎等》㉕*）
#  康熙十二年修《青州府志》第20卷载，万历四十三年（1615年），山东青州府推官[[:w:黄槐开|黄槐开]]在一件申文中说：“自古饥年，止闻道殣相望与易子而食、析骸而爨耳。今屠割活人以供朝夕，父子不问矣，夫妇不问矣，兄弟不问矣。剖腹剜心，支解作脍，且以人心味为美，小儿味尤为美。甚有鬻人肉于市，每斤价钱六文者；有腌人肉于家，以备不时之需者；有割人头用火烧熟而吮其脑者；有饿方倒而众刀攒割立尽者；亦有割肉将尽而眼瞪瞪视人者。间有为人所诃禁，辄应曰：&quot;我不食人，人将食我。&quot;愚民恬不为怪，有司法无所施。枭獍在途，天地昼晦。”
#  1616年，万历四十四年：四十四年，山东饥甚，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;今春以来，天鼓两震于晋地，流星昼陨于清丰，地震二十八，天火九，石首雨菽，河内女妖，辽东兵端吐火，即春秋二百四十年间，未有稠于今日者。且山东大昆，人相食，黄河水稽天。（《明史·卷二百五十七·列传第一百四十五·张鹤鸣等》㉕）&lt;/p&gt;&lt;p&gt;“以山东大饥，致母食死儿，夫食死妻，再振之。”（《中外历史年表》）&lt;/p&gt;
#  萬曆四十五年（1617年）連兩年山東大饑，蔡州有人肉市。“中州兄弟两无子，去山东买妾，遇二女，自称姑嫂，骗兄弟往。兄得小姑。小姑私语之曰：汝弟已为我嫂制成肉羹矣。兄急往视，弟头尚扔炕下。兄急诉之县，抵嫂于罪，兄带小姑去。”（《[[:w:棗林杂俎|棗林杂俎]]》）
#  近日福建抽稅太監高采謬聽方士言：食小兒腦千餘，其陽道可復生如故。乃遍買童稚潛殺之。久而事彰聞，民間無肯鬻者，則令人遍往他所盜至送入，四方失兒者無算，遂至激變掣回。此等俱飛天夜叉化身也。&lt;ref&gt;[[s:萬曆野獲編/卷28#食人]]&lt;/ref&gt;
#  约1621年，天启初：天启初，奢崇明反，率众薄城。（董）尽伦偕知州翁登彦固守。贼遣使说降，尽伦大怒，手刃贼使，抉其睛啖之，屡挫贼锋，城获全。（《明史·卷二百九十·列传第一百七十八·忠义二》㉕*）
#  1622年，天启二年：万化亦率苗仲九股陷龙里，遂围贵阳，自称罗甸王，时天启二年二月也。……外援既绝，攻益急，城中粮尽，人相食，而拒守不遗余力。（《明史·卷三百十六·列传第二百四·贵州土司》㉕*）&lt;p&gt;方官廪之告竭也，米升直二十金。食糠核草木败革皆尽，食死人肉，后乃生食人，至亲属相啖。彦方、运清部卒公屠人市肆，斤易银一两。枟尽焚书籍冠服，预戒家人，急则自尽，皆授以刀缳。城中户十万，围困三百日，仅存者千余人。（《明史·卷二百四十九·列传第一百三十七·朱燮元等》㉕）&lt;/p&gt;
#  1627年，清皇太极之天聪元年，天启七年：（清）国中大饥，斗米价银八两（天启时金一两合銀十两），人有相食者。国中银两虽多，无外贸易，是以银贱而诸物腾贵。良马银300两，牛一银百两，蟒缎一，银百五十两，布一匹，银九两。（《清太宗实录卷三》）
#  “天启辛酉，延安、庆阳、平凉旱，岁大饥。东事孔棘，有司惟顾军兴，征督如故，民不能供，道殣相望。或群职富者粟，惧捕诛，始聚为盗。盗起，饥益甚，连年赤地，斗米千钱不能得，人相食，从乱如归。饥民为贼由此而始。”&lt;ref&gt;《怀陵流寇始终录》，卷1，1页。&lt;/ref&gt;
#  1629年，崇禎二年，殺[[:w:袁崇煥|袁崇煥]]。[[:w:張岱|張岱]]《石匱書後集》：“（袁崇煥）遂於鎮撫司綁發西市，寸寸臠割之。割肉一塊，京師百姓從劊子手爭取生啖之。劊子亂撲，百姓以錢爭買其肉，頃刻立荊開腔出其腸胃，百姓群起搶之，得其一節者，和燒酒生嚙，血流齒頰間，猶唾地罵不已。拾得其骨者，以刀斧碎磔之，骨肉俱盡，止剩一首，傳視九邊。”，“时百姓怨恨，争啖其肉，皮骨已尽，心肺之间犹叫声不绝，半日而止，所谓活剐者也……百姓将银一钱，买肉一块，如手指大，噉之。食时必骂一声，须臾崇焕肉悉卖尽。”（[[:w:计六奇|计六奇]]：《[[:w:明季北略|明季北略]]》卷五）
#  1633年，崇祯六年：（陈）三接，文水人。举崇祯六年乡试，知河间县。岁旱饥，人相食。（《明史·卷二百九十一·列传第一百七十九·忠义三》㉕*）
#  1634年，崇祯七年：七年，京师饥，御史龚廷献绘《饥民图》以进。太原大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;七年，西北大旱，秦、晋人相食，（吴甘来）疏请发粟以振。（《明史·卷二百六十六·列传第一百五十四·马世奇等》㉕）&lt;/p&gt;
#  1636年，崇祯九年：山西大饥，人相食。（《明史·卷二十三·本纪第二十三·庄烈帝一》㉕*）
#   1637年，崇祯十年：十年浙江大饥，父子、兄弟、夫妻相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  崇禎十二年（1639年）[[:w:鄭鄤|鄭鄤]]以「杖母、姦妹」罪被磔死。《[[:w:明季北略|明季北略]]》记载鄭鄤被凌迟三千六百刀後，为“都人士”药用：“炮声响后，人皆跻足引领，顿高尺许，拥挤之极……归途所见，买生肉为疮疥药料者，遍长安市。二十年前之文章气节，功名显宦，竟与参术甘皮同奏肤功，亦大奇也。”
#  1639年，崇祯十二年：十二年，两畿、山东、山西、陕西、江西饥。河南大饥，人相食，卢氏、嵩、伊阳三县尤甚。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1640年，崇禎十三年，全國有123州縣發生“人相食”，98州縣蝗災。{{Citation needed}}&lt;p&gt;是年，两畿、山东、河南、山、陕旱蝗，人相食。（《明史·卷二十四·本纪第二十四·庄烈帝二》㉕*）&lt;/p&gt;&lt;p&gt;关河大旱，人相食，土寇蜂起，陕西窦开远、河南李际遇为之魁，饥民从之，所在告警。（《明史·卷二百五十二·列传第一百四十·杨嗣昌等》㉕）&lt;/p&gt;&lt;p&gt;十三年，北畿、山东、河南、陕西、山西、浙江、三吴皆饥。自淮而北至畿南，树皮食尽，发瘗胔（坟墓里的尸体）以食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕）&lt;/p&gt;
#  1641年，崇祯十四年：德州斗米千钱，父子相食，行人断绝。大盗滋矣。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;/p&gt;&lt;p&gt;及崇祯时，常洵地近属尊，朝廷尊礼之。常洵日闭阁饮醇酒，所好惟妇女倡乐。秦中流贼起，河南大旱蝗，人相食，民间藉藉，谓先帝耗天下以肥王，洛阳富于大内。（《明史·卷一百二十·列传第八·诸王五》㉕）&lt;/p&gt;&lt;p&gt;芳奕，慷慨负智略，与秉衡同举于乡，为昌乐知县。解官归，岁大歉，人相食，倾橐济之。（《明史·卷二百九十三·列传第一百八十一·忠义五》㉕）&lt;/p&gt;&lt;p&gt;十四年（左懋第）督催漕运，道中驰疏言：“臣自静海抵临清，见人民饥死者三，疫死者三，为盗者四。米石银二十四两，人死取以食，惟圣明垂念。”（《明史·卷二百七十五·列传第一百六十三·张慎言等》㉕）&lt;/p&gt; 崇禎十四年（1641年），「浙江大旱，飛蝗蔽天，食草根幾盡，人饑且疫」。崇祯十四年二月，时山东荒旱，寇盗益炽，徐德（南端到北端）数千里-{}-白骨纵横，父子相食，人迹断绝。（彭贻孙《平寇志》）
#  1641年，崇祯十四年：（九月）十一日，秦师食尽，宗龙杀马骡以享军。明日，营中马骡尽，杀贼取其尸分啖之。（《明史·卷二百六十二·列传第一百五十·傅宗龙等》㉕*）
#  明朝末年，四川大饑，“蜀大飢，人相食。先是丙戌、丁亥，連歲干涸，至是彌甚。赤地千里，糲米一斗價二十金，養麥一斗價七八金，久之亦無賣者篙芹木葉，取食殆盡。時有裹珍珠二昇，易一面不得而殆：有持數百金，買一飽不得而死。於是人皆相食，道路飢殍，剝取殆盡。無所得，父子、兄弟、夫妻，轉相賊殺。”（清·彭遵泅《蜀碧》卷四）
#  「庚辰山西大饑，人相食，剖心，其竅多寡不等。或無竅，或五六，其二、三竅為多，心大小各異。」（《[[:w:棗林雜俎|棗林雜俎]]·和集》）
#  明朝崇禎末年，河南和山東發生饑荒和蝗災，可以吃的東西都已經吃完，唯一剩下的可以吃的就只有人，於是便有了公開的人肉市場，其販賣的乃是活生生的人，稱之曰“[[:w:菜人|菜人]]”。[[:w:紀昀|紀昀]]《[[:w:閱微草堂筆記|閱微草堂筆記]]》有這樣的記載：“婦女幼孩，反接鬻於市，謂之菜人”。&lt;ref&gt;{{cite wikisource |title=《閱微草堂筆記》 |wslink=閱微草堂筆記 |chapter=卷2 |author=紀昀 | authorlink=紀昀}}&lt;/ref&gt;而在[[:w:屈大均|屈大均]]創作的一首七言古詩《[[s:菜人哀|菜人哀]]》，內容即以第一視角描述一對夫妻在崇禎末年，一位丈夫因過於飢餓，將妻子賣予一家屠戶成為“菜人”。
#  《陕西通志》第86卷载有明朝崇祯年间[[:w:马懋才|马懋才]]的《备陈灾变疏》，疏中写道：“臣乡延安府，自去岁一年无雨，草木枯焦。八、九月间，民争采山间蓬草而食，其粒类糠皮，其味苦而涩，食之仅可延以不死。至十月以后而蓬尽矣；则剥树皮而食。诸树惟榆树差善，杂他树皮以为食，亦可稍缓其死。殆年终而树皮又尽矣，则又掘山中石块而食。甘石名青叶，味腥而腻，少食辄饱，不数日则腹胀下坠而死。民有不甘于食石以死者始相聚为盗，而一、二稍有积贮之民遂为所劫，而抢掠无遗矣。有司亦不能禁治。间有获者亦恬不知畏；且曰：“死于饥与死于盗等耳，与其坐而饥死，何若为盗而死，犹得为饱鬼也。”
#  [[:w:計六奇|計六奇]]說：“天降奇荒，所以资自成也！”&lt;ref&gt;{{cite wikisource |title=《明季北略》 |wslink=明季北略 |chapter=卷05 |author=計六奇|authorlink=計六奇}}&lt;/ref&gt;。 
#  崇禎十四年（1641年）二月，[[:w:李自成|李自成]]攻陷洛陽，殺重達三百六十多斤的福王[[:w:朱常洵|朱常洵]]，用他的肉和皇家園林裡的[[:w:梅花鹿|梅花鹿]]一同烹煮，在洛陽西關周公廟舉行宴會，賜給部下食用，名曰“福祿宴”。&lt;ref&gt;《明季北略·卷十七》：王体肥，重三百余筋，贼置酒大会，以王为菹，杂鹿肉食之，号福禄酒。&lt;/ref&gt;
#  约1644年，顺治二年：（刘）泽清颇涉文艺，好吟咏。尝召客饮酒唱和。幕中蓄两猿，以名呼之即至。一日，宴其故人子，酌酒金瓯中，瓯可容三升许，呼猿捧酒跪送客。猿狰狞甚，客战掉，逡巡不敢取。泽清笑曰：“君怖耶？”命取囚扑死阶下，剜其脑及心肝，置瓯中，和酒，付猿捧之前。饮酹，颜色自若。其凶忍多此类。（《明史·卷二百七十三·列传第一百六十一·左良玉等》㉕*）
#   明末：中原盗起十余年，所在荼毒，督抚莫能办，率倡抚议，苟且幸无事，盗且服且叛。而河南比年大旱蝗，人相食，民益蜂起为盗。（《清史稿·卷五百·列传二百八十七·遗逸一》㉕*）
#  时有将军安氵侃者，一岁丧母，事其父以孝闻。父病革，刲臂为汤饮父，父良已。（《明史·卷一百十六·列传第四·诸王》㉕*）
#  襄陵王冲秌，宪王第二子，有至性。母病，刲股和药，病良已。（《明史·卷一百十八·列传第六·诸王三》㉕*）
# （襄陵王冲秌之）子范址服其教，母荆罹危疾，亦刲股进之，愈。（《明史·卷一百十八·列传第六·诸王三》㉕*）
#  刘铉，字宗器，长洲人。生弥月而孤。及长，刲股疗母疾。母卒，哀毁，以孝闻。（《明史·卷一百六十三·列传第五十一·李时勉等》㉕*）
# （孙）祖寿初守固关，遘危疾，妻张氏割臂以疗，绝饮食者七日。祖寿生，而张氏旋死，遂终身不近妇人。（《明史·卷二百七十一·列传第一百五十九·贺世贤》㉕*）
#  朱鉴，字用明，晋江人。童时刲股疗父疾。举乡试，授蒲圻教谕。（《明史·卷一百七十二·列传第六十·罗亨信等》㉕*）
#  储巏，字静夫，泰州人。九岁能属文。母疾，刲股疗之，卒不起。（《明史·卷二百八十六·列传第一百七十四·文苑二》㉕*）
#  许琰，字玉仲，吴县人。幼有至性，尝刲臂疗父疾。（《明史·卷二百九十五·列传第一百八十三·忠义七》㉕*）
#  沈德四，直隶华亭人。祖母疾，刲股疗之愈。己而祖父疾，又刲肝作汤进之，亦愈。洪武二十六年被旌。寻授太常赞礼郎。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  上元姚金玉、昌平王德儿亦以刲肝愈母疾，与德四同旌。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  至二十七年九月，山东守臣言：“日照民江伯儿，母疾，割肋肉以疗，不愈。祷岱岳神，母疾瘳，愿杀子以祀。已果瘳，竟杀其三岁儿。”帝大怒曰：“父子天伦至重。《礼》父服长子三年。今小民无知，灭伦害理，亟宜治罪。”遂逮伯儿，仗之百，遣戍海南。因命议旌表例。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  永乐间，江阴卫卒徐佛保等复以割股被旌。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  夏子孝，字以忠，桐城人。六岁失母，哀哭如成人。九岁父得危疾，祷天地，刲股六寸许，调羹以进，父食之顿愈。翌日，子孝痛创，父诘其故，始知之。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  金子良亦有孝行，父病，刲股为羹以进，旋愈。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  唐俨，全州诸生也。父荫，郴州知州，归老得危疾。俨年十二，潜割臂肉进之，疾良已。及父殁，哀毁如成人。其后游学于外，嫡母寝疾。俨妻邓氏年十八，奋曰：“吾妇人，安知汤药。昔夫子以臂肉疗吾舅，吾独不能疗吾姑哉？”于是割胁肉以进，姑疾亦愈。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  刘孝妇，新乐韩太初妻。……刘事姑谨，姑道病，刺血和药以进。……及姑疾笃，刲肉食之，少苏，逾月而卒，殡之舍侧。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  程氏，扬州胡尚絅妻。尚絅婴危疾，妇刲腕肉啖之，不能咽而卒。妇号恸不食二日。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  杨泰奴，仁和杨得安女。许嫁未行。天顺四年，母疫病不愈。泰奴三割胸肉食母，不效。一日薄幕，剖胸取肝一片，昏仆良久。及苏，以衣裹创，手和粥以进，母遂愈。母宿有膝挛疾，亦愈。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  后有张氏，仪真周祥妻。姑病，医百方不效。一方士至其门曰：“人肝可疗。”张割左胁下，得膜如絮，以手探之没腕，取肝二寸许，无少痛，作羹以进姑，病遂瘳。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  李孝妇，临武人，名中姑，适江西桂廷凤。姑邓患痰疾，将不起，妇涕泣忧悼。闻有言乳肉可疗者，心识之。一日，煮药，巘香祷灶神，自割一乳，昏仆于地，气已绝。廷凤呼药不至，出视，见血流满地，大惊呼救，倾骇城市，邑长佐皆诣其庐，命亟治。俄有僧踵门曰：“以室中蕲艾傅之，即愈。”如其言，果苏，比求僧不复见矣。乃取乳和药奉姑，姑竟获全。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  洪氏，怀宁章崇雅妻。崇雅早卒，洪守志十年。姑许，疾不能起，洪剜乳肉为羹而饮之，获愈，余肉投池中，不令人知。数日后，群鸭自水中衔出，鸣噪回翔，小童获以告姑。姑起视之，乳血犹淋漓也。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  倪氏，兴化陆鳌妻。性纯孝，舅早世，悯姑老，朝夕侍寝处，与夫睽异者十五年。姑鼻患疽垂毙，躬为吮治，不愈，乃夜焚香告天，割左臂肉以进，姑啖之愈。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  刘氏，张能信妻，太仆卿宪宠女，工部尚书九德妇也。性至孝，姑病十年，侍汤药不离侧。及病剧，举刀刲臂，侍婢惊持之。舅闻，嘱医言病不宜近腥腻，力止之。逾日，竟刲肉煮糜以进，则乃姑已不能食，乃大悔恨曰：“医绐我，使姑未鉴我心。”复刲肉寸许，恸哭奠箦前，将阖棺，取所奠置棺中曰：“妇不获复事我姑，以此肉伴姑侧，犹身事姑也。”乡人莫不称其孝。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  （颍）州又有台氏，诸生张云鹏妻。夫病，氏单衣蔬食，祷天愿代，割臂为糜以进。（《明史·卷三百三·列传第一百九十一·列女三》㉕*）

==清==
《清史稿》记载的割肉疗亲的事迹比二十五史以往各朝都多，但其实雍正有一段诏书不赞同割肉疗亲，朝廷的实际做法似乎是迫于民情不得已的情况下低调褒奖（“破格报可”），社会风气看来是称赞这种行为的。

* 雍正六年三月壬子，世宗谕曰：“……父母爱子，无所不至，若因己病而致其子割肝刲股以充饮馔、和汤药，纵其子无恙，父母未有不惊忧恻怛惨惕而不安者，况因此而伤生，岂父母所忍闻乎？父母有疾，固人子尽心竭力之时，傥能至诚纯孝，必且感天地、动鬼神，不必以惊世骇俗之为，著奇于日用伦常之外。……倘训谕之后，仍有不爱躯命，蹈于危亡者，朕亦不概加旌表，以成激烈轻生之习也。”（《清史稿·卷四百九十七·列传二百八十四·孝义一》）
* 清兴关外，俗纯朴，爱亲敬长，内悫而外严。既定鼎，礼教益备。定旌格，循明旧。亲存，奉侍竭其力；亲殁，善居丧，或庐于墓；亲远行，万里行求，或生还，或以丧归。友于兄弟，同居三五世以上，号义门，及诸义行，皆礼旌。亲病，刲股刳肝；亲丧，以身殉：皆以伤生有禁，有司以事闻，辄破格报可。所以教民者，若是其周其密也。国史承前例，撰次孝友传，亦颇及诸义行。（《清史稿·卷四百九十七·列传二百八十四·孝义一》）

历史记载中清朝的食人事件：

#  努尔哈赤时代：扬古利，舒穆禄氏，世居浑春。父郎柱，为库尔喀部长，率先附太祖，……扬古利手刃杀父者，割耳鼻生啖之，时年甫十四，太祖深异焉。（《清史稿·卷二百二十六·列传十三·扬古利等》㉕*）
#  清初：虞尔忘、尔雪，江南无锡人。国初江南多盗，尔忘、尔雪父罕卿董乡团，……罕卿死桥下矣。……知为盗杜息（所杀）。….. 比明，尔忘抱罕卿木主至，尔雪于其旁爇釜，尔忘取（杜）息舌，尔雪探心肝，且祭且啖，尔忘乃断（杜）息头。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  1627年，天聪元年，《太宗实录卷三》：“时国中大饥，斗米价银八两，人有相食者。国中银两虽多，无外贸易，是以银贱而诸物腾贵。良马，银三百两。牛一，银百两。蟒缎一，银百五十两。布匹一，银九两。盗贼繁兴，偷窃牛马，或行劫杀。于是诸臣入奏曰：盗贼若不按律严惩，恐不能止息。上恻然，谕曰：今岁国中因年饥乏食，致民不得已而为盗耳。缉获者，鞭而释之可也。遂下令，是岁谳狱，姑从宽典。仍大发帑金，散赈饥民。”
#  1631年，皇太极天聪四年：顷大凌河之役，城中人相食，明人犹死守，及援尽城降，而锦州、松、杏犹不下。（《清史稿·卷二·本纪二·太宗本纪一》㉕*）&lt;p&gt;旋有王世龙者，越城出降，言城中粮竭，商贾诸杂役多死，存者人相食，马毙殆尽。（《清史稿·卷二百三十四·列传二十一·孔有德等》㉕）&lt;/p&gt;&lt;p&gt;祖大壽疏奏：“被圍將及三月，城中食盡，殺人相食。”（《崇禎長編》卷五二）。&lt;/p&gt;&lt;p&gt;明大凌河城內，糧絕薪盡。軍士飢甚，殺其修城夫役及商賈平民為食，析骸而炊。又執軍士之羸弱者，殺而食之。（《清太宗實錄·卷十》）&lt;/p&gt;
#  1635年，皇太极天聪八年：先是，察哈尔林丹西奔图白特，其部众苦林丹暴虐，逗遛者什七八，食尽，杀人相食，屠劫不已，溃散四出。（《清史稿·卷二·本纪二·太宗本纪一》㉕*）
#  1645年，顺治二年：二年，耒（枣？）阳、襄阳、光化、宜城大饥，人相食。”（{{cite wikisource |title=《清史稿·卷44·志十九·災異五》 |wslink=清史稿/卷44 |author=趙爾巽|authorlink=趙爾巽}}㉕*）
#  1648年，顺治五年：五年春，广州、鹤庆（大理，洱海之北）嵩明（昆明市东北）大饥，人相食。”（{{cite wikisource |title=《清史稿·卷42·志十七·災異三》 |wslink=清史稿/卷42 |author=趙爾巽|authorlink=趙爾巽}}㉕*）
#  順治九年八月，漳州被圍半年，城中缺糧，一碗稀粥索價白銀四兩。居民以老鼠、麻雀、樹根、樹葉、水萍、紙張和皮革等物為食，餓死者不計其數，“城中人自相食，百姓十死其八，兵馬盡皆枵腹”&lt;ref&gt;《明清史料》丁編，第一本，第七十五頁《查明漳州解圍功次殘件》。&lt;/ref&gt;。
#  1654年，顺治十一年：顺治十一年，明将李定国攻新会，城守阅八月，食尽，杀人马为食。（《清史稿·卷五百十·列传二百九十七·列女三》㉕*）
#  顺治年間，“安邑知县鹿尽心者，得痿痺疾。有方士挟乩术，自称刘海蟾，教以食小儿脑即愈。鹿信之，辄以重价购小儿击杀食之，所杀伤甚众，而病不减。因复请于乩仙，复教以生食乃可愈。因更生凿小儿脑吸之。致死者不一，病竟不愈而死。事随彰闻，被害之家，共置方士于法。”&lt;ref&gt;[[:w:王士祯|王士祯]]：《池北偶谈·鹿尽心》&lt;/ref&gt;
#  康熙十八年（1679年），山东“终年不雨，大饥，人相食。”（乾隆《青城（即今高青）县-{}-志》卷10）
#  1681年，康熙二十年：诇知粮将罄，人相食，与诸将环而攻之。（吴）世璠众内乱，欲擒世璠以降，世璠自杀。（《清史稿·卷二百五十四·列传四十一·赉塔等》㉕*）
#  1698年，康熙三十七年春：三十七年春，平定、乐平大饥，人相食。”（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1703年，康熙四十二年：永年（邯郸东北）、东明（大名府之南部，山东曹州西）饥。秋：沛县、亳州、东阿、曲阜、蒲县（属隰州，非蒲城县）、滕县大饥。冬，汶上、沂州、莒州、兖州、东昌、郓城大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1704年，康熙四十三年：四十三年春，泰安大饥，人相食，死者枕藉。肥城，东平大饥，人相食。武定（惠民）、滨州（武定东）、商河（武定西南）、阳信（武定北）、利津、沾化饥；兖州、登州大饥，民死大半，至食屋草；昌邑、即墨、掖县、高密、膠州大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#   1785年，乾隆五十年：秋，寿光、昌乐、安丘、诸城大饥，父子相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1786年，乾隆五十一年：五十一年春，山东各府、州、县大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）&lt;p&gt;《病榻梦痕录》卷上乾隆五十一年（１７８６）条记载了苏皖鲁等地的灾情，时灾民卖妻鬻子，“流丐载道”，“尸横道路”，尸体“埋于土，辄被人刨发，刮肉而啖”。&lt;/p&gt;
#  1801，嘉庆六年: 罗思举，字天鹏，四川东乡人。……（嘉庆）六年，歼张世龙于铁溪河，……自是转战老林，饷不时至，煮马鞯，啗贼肉以追贼。……尝酒酣袒身示人，战创斑斑，为父母刲股痕凡七，其忠孝盖出天性云。（《清史稿·卷三百四十七·列传一百三十四·杨遇春等》㉕*）
#  1832年，道光十二年：夏，紫阳大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1833年，道光十三年：夏，保康、郧县、房县饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1834年，道光十四年：十四年春，归州、兴山大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1847年，道光二十七年：二十七年，南乐饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1857年，咸丰七年：七年春，肥城、东平大饥，死者枕藉；鱼台、日照、临朐亦饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1863年，同治二年，[[:w:石達開|石達開]]的軍隊為[[:w:大渡河|大渡河]]的涨水所阻，當時石部全軍已是“覓食無所得，有相殺噬人肉者”。（许亮儒遗著《擒石野史》）
#  [[:w:陈康祺|陈康祺]]《郎潜纪闻二笔》记载“同治三、四年，皖南到处食人，人肉始买三十文一斤，后增至一百二十文一斤，句容、二溧，八十文一斤，惨矣。”
#  同治三年（1864年），皖南人相食，人肉價格大漲。《曾国藩日记》同治三年四月廿二日记载：“皖南到处食人，人肉始卖三十文一斤，近闻增至百二十文一斤，句容、二溧八十文一斤。”《曾國藩日記》又記載：“[[:w:太平天国|洪楊]]之亂，[[:w:江蘇|江蘇]]人肉賣九十文一斤，漲到一百三十文錢一斤。”
#  1866年，同治五年：五年，兰州饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1867年，同治六年：五年，（穆图善）收灵州。……明年，署陕甘总督，值岁大饥，人相食。（《清史稿·卷四百五十四·列传二百四十一·刘锦棠》㉕*）
#  1868年，同治七年：七年春，即墨、孝义厅、蓝田、沔县饥。夏，泾州大饥，人相食。《清史稿·卷四十四·志十九·灾异五》㉕*）&lt;p&gt;时庆阳大饥，人相食。（《清史稿·卷四百五十四·列传二百四十一·刘锦棠等》㉕）&lt;/p&gt;&lt;p&gt; 同治七年（1868年），[[:w:定西|定西]]、[[:w:通渭|通渭]]大旱，時逢戰亂，瘟疫並起，人相食。{{Citation needed|Date=January 2025}}&lt;/p&gt;
#  1877年，光绪三年：是岁，山、陕大旱，人相食。（《清史稿·卷二十三·本纪二十三·德宗本纪一》㉕*）&lt;p&gt;丁戊奇荒是中国华北地区发生于清朝光绪元年（1875年）至四年（1878年）之间的一场罕见的特大旱灾饥荒。灾害波及山西、直隶、陕西、河南、山东、甘肃等好几个省份，“饿殍载途，白骨盈野”，饿死的人竟达一千万以上，逃亡两千万以上。随著灾情的发展，可食之物的罄尽，“人食人”的惨剧发生了。大旱的第三年（1877年）冬天，重灾区山西，到处都有人食人现象。吃人肉、卖人肉者，比比皆是。有活人吃死人肉的，还有将老人或孩子活杀吃的……无情旱魔，把灾区变成了人间地狱! 在河南，侥幸活下来的饥民大多奄奄一息，“既无可食之肉，又无割人之力”，一些气息犹存的灾民，倒地之后即为饿犬残食。{{Citation needed|Date=January 2025}}《申报》1877年12月7日载：“今岁豫省之灾，亦不减于山右，……灾黎数百万，几有易子析骸之惨”&lt;/p&gt;
#  1900年，光绪二十六年：二十六年，两宫西狩，关中大饥，人相食，（唐）锡晋醵金四十万往赈，历二州八县，艰困不少阻。（《清史稿·卷四百五十二·列传二百三十九·洪汝奎等》㉕*）
#  1910年，宣统二年十二月：是月，江、淮饥，人相食。东三省疫。（《清史稿·卷二十五·本纪二十五·宣统皇帝本纪》㉕*）
#  1911年，宣统三年：钟麟同，字建堂，山东济宁州人。威海武备学堂毕业。……宣统三年九月初九日，七十三标兵变，夜半，自北校场入城。……以手枪自击而仆，变军碎其尸，剖心啖之。上闻，有“忠骸支解，惨不忍闻”之谕，谥忠壮。（《清史稿·卷四百九十六·列传二百八十三·忠义十》㉕*）
#  光熙，本名惠熙，字亮臣。少从盛昱游，励学。钟琦遘危疾，尝刲股和药以进。（《清史稿·卷四百六十九·列传二百五十六·恩铭等》㉕*）
#  礼堂，字和贵。事亲孝。父继宏，久疟，冬月畏火，礼堂潜以身温被。居丧如礼，笑不见齿。母遘危疾，刲股合药，私祷于神，减齿以延亲寿。（《清史稿·卷四百八十一·列传二百六十八·儒林二》㉕*）
#  宋大樽，字左彝，仁和人。弱岁，刲股愈母疾，让产其弟。（《清史稿·卷四百八十五·列传二百七十二·文苑二》㉕*）
#  潘德舆，字四农，山阳人。年五六岁，母病不食，亦不食。父咯血，刲臂肉和药进，父察其色动，泣曰：“固知儿有是也！”（《清史稿·卷四百八十六·列传二百七十三·文苑三》㉕*）
#  曾艾，字虎卿，湖南新化人。尝割左臂疗父疾。（《清史稿·卷四百八十九·列传二百七十六·忠义三》㉕*）
#  陈源兖，字岱云，湖南茶陵州人。道光十八年进士，改翰林，授编修，旋授江西吉安府。先是源兖妻易氏以源兖遘疾几殆，籥天原以身代，刲臂和药饮源兖，源兖以愈，易氏旋病卒。同乡公举孝妇，请旌于朝。（《清史稿·卷四百九十·列传二百七十七·忠义四》㉕*）
#  沈瀛，字士登，江苏吴县人。尝刲臂疗母疾。（《清史稿·卷四百九十六·列传二百八十三·忠义十》㉕*）
#  李盛山，福建罗源人。母病，割肝以救，伤重，卒。巡抚常赉疏请旌，下礼部，礼部议轻生愚孝，无旌表之例。雍正六年三月壬子，世宗谕曰：“……父母爱子，无所不至，若因己病而致其子割肝刲股以充饮馔、和汤药，纵其子无恙，父母未有不惊忧恻怛惨惕而不安者，况因此而伤生，岂父母所忍闻乎？父母有疾，固人子尽心竭力之时，傥能至诚纯孝，必且感天地、动鬼神，不必以惊世骇俗之为，著奇于日用伦常之外。……倘训谕之后，仍有不爱躯命，蹈于危亡者，朕亦不概加旌表，以成激烈轻生之习也。”盛山仍予旌表。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  吕斅孚，湖南永定人。父孟卿，贫，以客授自给。母病将殆，思肉食，斅孚方七岁，贷诸屠，屠不可，泣而归。闻母呻吟，益痛，内念股肉可啗母，取厨刀砺使利，割右股四寸许，授其女弟，方五岁，令就炉火炙以进。母疾良已，孟卿归，察斅孚足微跛，得其状，与母持以哭。斅孚曰：“毋然，儿固无所苦也。”……孟卿亦尝刲股愈父病，然斅孚割股时，初不知父有是事也。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  汪灏，江南休宁人。晨、日昂、日升，其弟也。父病咯血，灏年十六，割股和药进，良愈。后数年病足，晨割股炼为末，敷治亦愈。又数年复咯血，晨复割臂以疗。更数年，疾大作，灏复割臂，勿瘳。晨病，日昂泣曰：“吾兄割臂愈父，吾不能割以愈吾兄乎？”众尼之。懵且仆，匠治棺，日升持匠斧断指，血淋漓，调药以饮晨。有司表其门曰“一门四孝友”。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  觉罗色尔岱，满洲镶红旗人，德世库七世孙也。性笃孝。年十七，父病，医不效，乃割左臂为糜以进，病稍间，旋歾。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  康熙间，以割臂疗亲旌者，有翁杜、佟良，与色尔岱同时有克什布。翁杜，满洲镶白旗人；佟良，蒙古镶黄旗人：官防御。克什布，满洲镶红旗人，官三等侍卫。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  奚缉营，字圣辉，江苏宝山人。父士本，以孝旌。缉营幼读论语，至“父母之年，不可不知”，辄陨涕簌簌，师奇之，谓真孝子子也。母病，刲臂以疗。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  张三爱，江南歙县人。为人役。事母孝，母病，不能具药物。或谓之曰：“汝欲愈母病，盍刲肝？”三爱祷于丛祠，破腹，肝堕出，以右手劙肝，得指许，左手纳于腹，束以白麻。归以肝和羹饮母，母良愈，三爱创亦合。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  杨献恒，山东益都人。父加官，与济南杨开泰有隙，……开泰计必欲杀献恒，遣其子承恩至青州谋诸吏。献恒潜知之，持铁骨朵挟刃至所居。承恩方与吏耳语，伺其出，以铁骨朵击之，仆，急拔刀断其喉，又抉其睛啖之，诣县自陈，出所藏银为证。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  刘希向，江南山阳人。……父病，希向为割股，良愈。希向年六十，病噎，其子亦割股，刀钝，肉不决，剪之，乃下，然希向竟不瘳。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  县有嫠张陈氏，家贫，刲肉以奉姑，训予田十亩助其养。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  李孔昭，字光四，蓟州人。……崇祯十五年进士，……母病，刲股疗之。（《清史稿·卷五百一·列传二百八十八·遗逸二》㉕*）
#  萧学华妻贺，湖南安化人。贺父徙陕西，学华赘其家。年余，学华归省母，贺欲与俱，父不许，贺割股肉付夫以奉姑。姑適病，学华烹肉进，病良已。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  子日焜妻李，尝刲股愈母病，事祖姑及姑孝。姑病，割臂进，病目，舐以舌，良已。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  王钜妻施，钜，萧山人；施，富阳人。姑严，小不当意，辄呵斥，施屏息不敢声。姑病反胃甚，医以为不治，施刲股和药进，病良已，姑遇施如故。钜疾作，施视疾惫，病瘵卒，姑犹不善施。钜以刲股事告，视其尸，信，乃大恸曰：“吾负孝妇！”（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  陈文世妻刘，郧人。陈、刘皆农家，刘待年于陈。既婚，姑年七十二，病噎，刘割臂和药以进，疾少间；既而复作，不食已十日，垂尽矣。刘夜屏人，杀鸡誓于神，持小刀自劙其胸二寸许，出肝刲半，取布束创，以肝与鸡同瀹汤奉姑。姑久不言，忽曰：“汤香甚！”饮之竟，病良愈，刘亦旋平。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  林经妻陈，连江人，姑盲性卞，常臆妇藐己，陈断三指自明，姑为之悔。经病，刲股；经卒，以节终。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  林云铭妻蔡，云铭，闽人；……耿精忠反，下云铭狱，蔡忧之，呕血殷紫，女瑛佩剜臂肉入药，旋苏。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  崔龙见妻钱，名孟钿，字冠之，一字浣青。龙见，永济人；钱，武进人，侍郎维城女。九岁刲臂疗父疾。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  张茂信妻方，茂信，河津人；方仪徵人。方尝割股愈舅疾，舅与茂信皆卒，奉姑刘。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  （袁）进忠病，疡生于胫，（养）女刲股以疗，家人皆不知，而长女虐愈甚。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  王前洛聘妻林，潜山人。前洛病，林父饣鬼药，林潜刲股入药。前洛卒，固请奔丧，引刀誓不嫁。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  徐文经聘妻姚，名淑金，侯官人。文经卒，淑金屡求死，乃归于徐。贫，舅殁，姑疾作，刲股以疗。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  乔涌涛聘妻方，桐城人。涌涛卒，涌涛母丁亦病，方请于父母，归于乔。以姑病寒疾，亦薄其衣当风雪。刲股以进姑，病良已。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  袁绩懋妻左，绩懋见《忠义传》。左名锡璇，字芙江，阳湖人。事亲孝，父病，刲臂和药进。工诗善画，书法尤精，著有卷葹阁诗集。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  何其仁聘妻李，路南人。嘉庆十一年，年十六，未行。其仁及其父皆病笃，李割股畀叔母使送婿家。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  林国奎妻郑，闽人。国奎卒，有子二。郑将殉，姑诫以存孤，乃已。一子殇，遂自沉于江，渔者拯以还。姑疾，刲肝杂糜进，疾良已。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  吉山妻瓜尔佳氏，名惠兴，满洲人，杭州驻防。早寡，事姑谨，尝刲肱疗姑疾。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  王如义妻向，涪州人。幼能为诗文。如义，农家子，向恒劝之读。道光十六年，如义暴卒，姑喻之嫁，矢以死。舅病，为刲股。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  许会妻张，颍州人。姑姣而虐，恶张端谨不类，日诟且挞，张事姑益恭。姑病，刲股以疗，姑虐如故。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  安于磐妻朱、后妻田，于磐，贵州蛮夷司长官。初娶朱，事姑孝，姑病，刲股，卒。复娶田，于磐病，刲股。于磐卒，抚诸子成立。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  田养民妻杨，养民，朗溪司长官；杨，邑梅司人也。年十二，母病，刲股。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  伊嵩阿，拜都氏，满洲镶黄旗人；妻希光，钮祜禄氏，正白旗人，总督爱必达女也。伊嵩阿为大学士永贵从子，早卒。方病时，希光割股进，终不起，许以死。爱必达、永贵共喻之，誓毕婚嫁乃殉。为伊嵩阿弟娶，嫁女妹及二女，次女行之明日，自缢死。张遗诗于壁，略谓：“十载要盟，此日当报命。”乾隆四十六年三月事也。永贵疏闻，高宗为赋诗，旌其节。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  朱承宇妻曹，承宇，无锡人；曹，武进人：皆农家也。生二子、一女，而承宇死。承宇弟迫之嫁，曹以死拒。……哭于承宇墓，还，遂缢。……及敛，左臂创未合，盖承宇病时尝割臂也。父为讼于县，罪迫嫁者。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）

==中华民国==
1936年“3月1日万源曹家沟某家七人，饿毙四人；余三人气息奄奄，竟为逃荒饥民杀死，分割炙食无余。”{{cfn|许汉三|y=1985}}
1936年3月19日四川省报载：“北川县人肉每斤五百文。片口镇饥民张彭氏、何张氏、陈顺氏因饥饿难忍，挖掘死尸围食，被捕。”{{cfn|许汉三|y=1985}}

1936年四川《民间意识》杂志汇载四川各地吃人的消息：“松潘半边街居民陈氏，自杀其八岁的亲生女而食，食尽仍病饿而死。沿途数百里内，人血、白骨与饿死者，填满沟壑。”{{cfn|许汉三|y=1985}}

民國30年（1941）-民國32年（1943）河南省大旱，人相食。1942年河南省赈济会推选[[:w:杨一峰|杨一峰]]、[[:w:刘庄甫|刘庄甫]]、[[:w:任兆鲁|任兆鲁]]三人等赴[[:w:重庆|重庆]]，请国民党中央免除徵賦，蒋介石拒不接见。大公报主笔[[:w:王芸生|王芸生]]在1942年的一篇《看重庆，念中原》的社论中写道：“饿死的暴骨失肉，逃亡的扶老携幼，妻离子散，挤人丛，挨棍打，未必能够得到赈济委员会的登记证。吃杂草的毒发而死，吃干树皮的忍不住刺喉绞肠之苦。把妻女驮运到遥远的人肉市场，未必能够换到几斗粮食。”[[:w:冯小刚|冯小刚]]於2012年拍摄的电影《一九四二》讲的正是这段时期发生的故事。

1948年6月[[:w:國共內戰|國共內戰]]期間，[[:w:中共|中共]]将领[[:w:林彪|林彪]]進行[[:w:長春圍城|長春圍城]]，禁止糧食進城，國軍于是收集城內的糧食，造成很多人餓死街頭。10月21日，城內守軍[[:w:鄭洞國|鄭洞國]]投降。活過來的人說，「就喝死人腦瓜殼裡的水，都是蛆。就這麼熬著，盼著，盼開卡子放人。就那麼幾步遠，就那麼瞅著，等人家一句話放生。卡子上天天宣傳，說誰有槍就放誰出去。真有有槍的，真放，交上去就放人。每天都有，都是有錢人，在城裡買了準備好的，都是手槍。咱不知道。就是知道，哪有錢買呀！」參加圍城的中共官兵說：「在外邊就聽說城裡餓死多少人，還不覺怎麼的。從死人堆裡爬出多少回了，見多了，心腸硬了，不在乎了。可進城一看那樣子就震驚了，不少人就流淚了。」&lt;ref&gt;张正隆：《雪白血红》&lt;/ref&gt;

==中華人民共和國==

=== 三年大跃进时期 ===
1959年－1961年「[[:w:大跃进|大躍進]]」期間，中國大陸發生“[[:w:三年困难时期|三年大饑荒]]”，据各方估计共造成1500万-5500万[[:w:非正常死亡|非正常死亡]]&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title=The Institutional Causes of China's Great Famine, 1959–1961|author=|url=https://www0.gsb.columbia.edu/faculty/pyared/papers/famines.pdf|first1=XIN|last2=QIAN|first2=NANCY|date=2015-01|journal=Review of Economic Studies|issue=4|doi=10.1093/restud/rdv016|others=|year=|volume=82|page=|pages=1568–1611|pmid=|last3=YARED|first3=PIERRE|archive-date=2019-09-06|url-status=|via=|last1=MENG|archive-url=https://web.archive.org/web/20190906163322/https://www0.gsb.columbia.edu/faculty/pyared/papers/famines.pdf|dead-url=no}}&lt;/ref&gt;&lt;ref name=&quot;:29&quot;&gt;{{Cite web|title=西方学术界的大跃进饥荒研究|url=http://ww2.usc.cuhk.edu.hk/PaperCollection/webmanager/wkfiles/2012/201503_38_paper.pdf|author=陈意新|date=2015-01|format=|work=[[:w:香港中文大学|香港中文大学]]|publisher=《江苏大学学报》|language=zh|archive-url=https://web.archive.org/web/20210517052743/http://ww2.usc.cuhk.edu.hk/PaperCollection/webmanager/wkfiles/2012/201503_38_paper.pdf|archive-date=2021-05-17|dead-url=no}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|title=SITES OF HORROR: MAO'S GREAT FAMINE [with Response]|author=Felix Wemheuer|url=http://www.jstor.org/stable/41262812|date=2011|journal=The China Journal|issue=66|doi=|others=|year=|editor-last=Dikötter|editor-first=Frank|volume=|page=|pages=155–164|issn=1324-9347|pmid=|archive-url=https://web.archive.org/web/20200727141524/https://www.jstor.org/stable/41262812|archive-date=2020-07-27|dead-url=no}}&lt;/ref&gt;。餓殍遍野，到處都有餓死倒斃在路邊的人，有些地方甚至出現吃人肉的現象。[[:w:楊繼繩|杨继绳]]所著的《[[:w:墓碑 (书籍)|墓碑]]》一書援引梁志遠的《關於「特種案件」的匯報——安徽亳縣人吃人見聞錄》記載指人吃人並不是個別現象：“其面積之廣，數量之多，時間之長，實屬世人罕見”{{cfn|楊繼繩|y=2008|p=274}}。

1960年春，吃人肉情況不斷發生，人肉的交易市場也隨之出現在城郊、集鎮、農民擺攤等{{cfn|楊繼繩|y=2008|p=278}}。三年大饑荒的[[:w:口述歷史|口述歷史]]《[[:w:尋找大饑荒倖存者|尋找大饑荒倖存者]]》记载了四十九起人吃人事件&lt;ref name=&quot;rfa&quot;&gt;{{Cite news|url=https://www.rfa.org/mandarin/pinglun/wenyitiandi-cite/yujie-01082014125845.html|title=为当代中国修筑一面“哭墙”--依娃《寻找大饥荒幸存者》|publisher=[[:w:自由亚洲电台|自由亚洲电台]]|date=2014-01-08|archive-url=https://web.archive.org/web/20210722001314/https://www.rfa.org/mandarin/pinglun/wenyitiandi-cite/yujie-01082014125845.html|archive-date=2020-07-22|dead-url=no|language=zh|author=余杰|authorlink=余杰}}&lt;/ref&gt;。人吃人事件在[[:w:四川|四川]]、[[:w:甘肅|甘肅]]、[[:w:青海|青海]]、[[:w:西藏|西藏]]、[[:w:陝西|陝西]]、[[:w:寧夏|寧夏]]、[[:w:河北|河北]]、[[:w:遼寧|遼寧]]皆有耳聞，幾乎遍及全國{{cfn|貝克|y=2005}}。據作家[[:w:沙青|沙青]]的[[:w:报告文学|報告文學]]記載：「有一戶農家，吃得只剩了父親和一男一女兩個孩子。一天，父親將女兒趕出門去，等女孩回家時，弟弟不見了，鍋裡浮著一層白花花油乎乎的東西，灶邊扔著一具骨頭。幾天之後，父親又往鍋裡添水，然後招呼女兒過去。女孩嚇得躲在門外大哭，哀求道：『爸爸，別吃我，我給你摟草、燒火，吃了我沒人給你做活。』」&lt;ref&gt;{{Cite web|title=依稀大地湾——大饥荒年代|url=https://boxun.com/news/gb/z_special/2004/12/200412281348.shtml?__cf_chl_jschl_tk__=pmd_cf65954eb189551663c797db8d490efde1f84d97-1626912600-0-gqNtZGzNAg2jcnBszQti|author=沙青|date=2004-12-28|publisher=[[:w:博讯|博讯]]|language=zh|archive-url=https://web.archive.org/web/20080822033646/http://www.peacehall.com/news/gb/z_special/2004/12/200412281348.shtml|archive-date=2008-08-22|dead-url=no}}&lt;/ref&gt;

* '''四川'''：《[[:w:中國大饑荒，1958-1962|中國大饑荒，1958-1962]]》引用的中國官方檔案中有吃人記載，如在[[:w:四川省|四川省]][[:w:石柱土家族自治縣|石柱土家族自治縣]]的桥头区，老妇人罗文秀是第一个开始吃人肉的人。在家人一家七口全部死去后，罗文秀把三岁女童马发慧的尸体挖出来。她把小女孩儿的肉割下来，用辣椒调味，然后蒸熟吃掉&lt;ref name=&quot;紐約時報&quot;&gt;{{cite news|url=http://cn.nytimes.com/china/20120917/c17famine/|title=記錄大饑荒人相食的慘劇|publisher=《[[:w:紐約時報|紐約時報]]》|date=2012年9月17日|archive-date=2013年10月23日|archive-url=https://web.archive.org/web/20131023013637/http://cn.nytimes.com/china/20120917/c17famine/|dead-url=no|author=DIDI KIRSTEN TATLOW|language=zh}}&lt;/ref&gt;。另一份1961年1月27日的文件，讲述了一个四川母亲用毛巾勒死了自己五岁大的儿子，“吃了四顿”。调查者王德明写道，“这样令人震惊的可怕事件远非只有这一起。”&lt;ref name=&quot;紐約時報&quot; /&gt;

* '''河南'''：1959年10月至1960年4月，[[:w:信阳事件|信陽事件]]，[[:w:商丘|商丘]]、[[:w:開封|開封]]餓得人身浮腫，吃樹皮，餓死100萬（到數百萬）人口，時諺：“人吃人，狗吃狗，老鼠餓得啃磚頭。”“信陽五里店村一個14、15歲的小女孩，将4、5歲的弟弟殺死煮了吃了，因爲父母都餓死了，只剩下這兩個孩子，女孩餓得不行，就吃弟弟。”{{cfn|楊繼繩|y=2008}} 河南省[[:w:固始县|固始縣]]官方記載有二百例人吃人事件，縣委以“破壞屍體”為名，逮捕群眾{{cfn|貝克|y=2005|p=180|url=https://books.google.com/books?id=hjpdAAAAIAAJ&amp;q=固始縣+二百}}。鹿邑、夏邑、虞城、永城等县共发现吃死人肉的情况20多起。据中央工作组魏震报告，鹿邑县从1959年10月到1960年11月，发现人吃人的事件6起。马庄公社马庄大队庞王庄18岁女子王玉娥于1960年4月19日将堂弟弟5岁的王怀郎溺死煮食，怀郎14岁的亲姐姐小朋也因饥饿吃了弟弟的肉。&lt;ref&gt;{{cite news |title=[杨继绳]《墓碑》――中国六十年代大饥荒纪实. |url=http://|publisher=第54頁 |accessdate=2022-03-23}}&lt;/ref&gt;

* '''甘肃'''：[[:w:通渭县|通渭縣]]，1958年全縣糧食實產8300多萬斤，虛報1.8億斤。人口大量死亡；有人回憶“1959年11月到臘月，死的人多。老百姓一想那事就要流淚。餓死老人家的，餓死婆娘的，日子過得糊裡糊塗。把人煮了吃，肉割來煮了吃……人甚麼也不想，甚麼也不怕，就想吃，想活。把娃娃、自己的娃娃吃下的，也有；把外面逃到村上的人殺了吃的，也有。吃下自己娃娃的，浮腫，中毒，不像人樣子。有的病死了，也有救下的。吃了娃娃心裡慘的，吃過就後悔了，自己恨自己。在村子里住不下去，沒人理他，嫌他臟。”（《50年代末大飢荒驚人記實》）
* '''青海'''：人吃人事件110多起，漢東公社楊家灘生產隊的婦女竟吃了9個小孩&lt;ref&gt;武文軍:《餓魂祭:中國六十年代饑荒考》，蘭州學刊2005年專輯，蘭州社會科學院主編，p110-110&lt;/ref&gt;。
* '''湖南'''：据余习广《吃人饿鬼：[[:w:刘家远惨杀亲子食子案|刘家远惨杀亲子食子案]]》記載，[[:w:湖南|湖南]][[:w:澧县|澧县]]如东公社男子刘家远，將自己儿子殺害後烹煮食用。刘家远也因食子而被處決&lt;ref&gt;{{cite news|title=毛泽东时代惨剧：三年大饥荒饥民十大奇吃|url=https://www.dwnews.com/%E4%B8%AD%E5%9B%BD/59674203/%E6%AF%9B%E6%B3%BD%E4%B8%9C%E6%97%B6%E4%BB%A3%E6%83%A8%E5%89%A7%E4%B8%89%E5%B9%B4%E5%A4%A7%E9%A5%A5%E8%8D%92%E9%A5%A5%E6%B0%91%E5%8D%81%E5%A4%A7%E5%A5%87%E5%90%83|publisher=[[:w:共识网|共识网]]|archive-date=2020-11-05|archive-url=https://web.archive.org/web/20201105165243/https://www.dwnews.com/%E4%B8%AD%E5%9B%BD/59674203/%E6%AF%9B%E6%B3%BD%E4%B8%9C%E6%97%B6%E4%BB%A3%E6%83%A8%E5%89%A7%E4%B8%89%E5%B9%B4%E5%A4%A7%E9%A5%A5%E8%8D%92%E9%A5%A5%E6%B0%91%E5%8D%81%E5%A4%A7%E5%A5%87%E5%90%83|dead-url=no|author=惠风（原作者：彭劲秀）|date=2014-03-11|language=zh|agency=[[:w:多維新聞|多維新聞]]}}&lt;/ref&gt;。
* '''安徽'''：作家[[:w:王立新 (1949年)|王立新]]1980年代曾赴[[:w:凤阳县|凤阳]]采访过，他在报告文学中写道：“梨园乡小岗生产队严俊冒告诉我：1960年，我们村附近有个死人塘，浮埋着许多饿死的人。为什么浮埋？饿得没力气呀，扔几锹土了事。说起来，对不起祖先，也对不起冤魂。人饿极了，什么事都干得出来。我的一位亲戚见人到死人塘割死人的腿肚子吃，她也去了。开始有点怕，后来惯了，顶黑去顶黑回。我问她：‘怎么能……？’她叹息道：‘饿极了。’”&lt;ref&gt;[[:w:李锐 (1917年)|李锐]]《大跃进亲历记》（南方出版社1999年版）&lt;/ref&gt;

=== 文化大革命时期 ===
{{main|:w:广西文革屠杀}}

[[:w:文化大革命|文化大革命]]時期（1966-1976年），[[:w:广西壮族自治区|广西壮族自治区]]除[[:w:广西文革屠杀|私刑、屠杀事件众多]]外，亦傳出多起食人事件&lt;ref name=&quot;:13&quot;&gt;{{Cite web|title=不反思“文革”的社会，就是个食人部落|url=http://history.people.com.cn/n/2013/0305/c200623-20680503.html|author=[[:w:张鸣 (学者)|张鸣]]|date=2013-03-05|format=|work=|publisher=《[[:w:中国青年报|中国青年报]]》|agency=[[:w:人民网|人民网]]|language=zh|archiveurl=https://web.archive.org/web/20200625141907/http://history.people.com.cn/n/2013/0305/c200623-20680503.html|archivedate=2020-06-25|dead-url=yes}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite web|title=我参与处理广西文革遗留问题|url=http://www.yhcqw.com/34/8938.html|accessdate=2019-11-29|author=晏乐斌|date=|format=|work=|publisher=《[[:w:炎黄春秋|炎黄春秋]]》|language=zh|archive-url=https://web.archive.org/web/20191207031844/http://www.yhcqw.com/34/8938.html|archive-date=2019-12-07|dead-url=yes}}&lt;/ref&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite web|title=广西文革中的吃人狂潮|url=http://www.cuhk.edu.hk/ics/21c/media/articles/c155-201605003.pdf|accessdate=|author=[[:w:宋永毅|宋永毅]]|date=|format=|publisher=[[:w:香港中文大学|香港中文大学]]|language=zh|archive-url=https://web.archive.org/web/20180127184237/http://www.cuhk.edu.hk/ics/21c/media/articles/c155-201605003.pdf|archive-date=2018-01-27|dead-url=no}}&lt;/ref&gt;。作家[[:w:鄭義 (作家)|鄭義]]曾在文革後赴廣西調查，于1993年出版《[[:w:红色纪念碑|红色纪念碑]]》一书，據他的統計廣西全省至少有一千人被食。紀錄片「文革廣西[[:w:武宣县|武宣縣]]紅衛兵吃人肉事件」評論称：“這些食人事件並不是因為飢荒，而是因為政治運動製造出來的仇恨心態&lt;ref&gt;{{Cite web |url=https://www.youtube.com/watch?v=vR2JhwcEM1A |title=文革廣西武宣縣紅衛兵吃人肉事件 |accessdate=2015-07-25 |archive-date=2016-03-16 |archive-url=https://web.archive.org/web/20160316105309/https://www.youtube.com/watch?v=vR2JhwcEM1A |dead-url=no }}&lt;/ref&gt;”。

其中人食人最厲害的地方之一是廣西[[:w:武宣县|武宣縣]]，官方调查发现至少38人被吃&lt;ref name=&quot;:0&quot; /&gt;，民间研究调查则发现有70余人&lt;ref name=&quot;:4&quot; /&gt;甚至上百人被吃&lt;ref name=&quot;:12&quot;&gt;{{Cite web|title=Chronology of Mass Killings during the Chinese Cultural Revolution (1966-1976)|url=https://www.sciencespo.fr/mass-violence-war-massacre-resistance/en/document/chronology-mass-killings-during-chinese-cultural-revolution-1966-1976|accessdate=|author=[[:w:宋永毅|宋永毅]]|date=2011-08-25|format=|publisher=[[:w:巴黎政治学院|巴黎政治学院]]（Sciences Po）|language=en|archive-url=https://web.archive.org/web/20190425062821/https://www.sciencespo.fr/mass-violence-war-massacre-resistance/en/document/chronology-mass-killings-during-chinese-cultural-revolution-1966-1976|archive-date=2019-04-25|dead-url=no}}&lt;/ref&gt;。武宣县“一女民兵因参与杀人坚定勇敢，且专吃男人生殖器而臭名远播，并因此入党做官，官至武宣县革委副主任。处遗时期中共中央书记处一天一个电话催问处理结果，并严厉责问：‘像这样的人，为什么还不赶快开除党籍？’但该副主任拒不承认专吃生殖器，只承认一起吃过人。最后的处理是开除党籍，撤销领导职务。现已调离武宣。”{{cfn|鄭義|y=1993|p=74-75|url=https://books.google.com/books?id=IJBxAAAAIAAJ&amp;q=武宣縣+副主任}}

== 参考文献 ==
=== 引用 ===
{{Reflist|30em}}

=== 来源 ===
{{refbegin}}
* 王永寬：《中國古代酷刑》
* [[:w:黃文雄 (作家)|黃文雄]]：《中國食人史》
* 黃粹涵：《中國食人史料鈔》
* {{cite book
|author=许汉三
|title=《黃炎培年谱》
|url=https://books.google.com/books?id=z2djAAAAIAAJ
|year=1985年
|publisher=文史资料出版社
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-26
|archive-url=https://web.archive.org/web/20210426094608/https://books.google.com/books?id=z2djAAAAIAAJ
}}
* {{cite book
|author=鄭義
|title=《紅色紀念碑》
|url=https://books.google.com/books?id=IJBxAAAAIAAJ
|year=1993年
|publisher=華視文化
|isbn=978-957-572-048-3
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-26
|archive-url=https://web.archive.org/web/20210426200250/https://books.google.com/books?id=IJBxAAAAIAAJ
}}
* {{cite book
|author=楊繼繩
|author-link=楊繼繩
|title=《墓碑——中國六十年代大饑荒紀實 上篇》
|url=https://books.google.com/books?id=GnglAQAAMAAJ
|year=2008年
|publisher=天地圖書
|isbn=978-988-211-909-3
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-19
|archive-url=https://web.archive.org/web/20210419003552/https://books.google.com/books?id=GnglAQAAMAAJ
}}
* {{cite book 
| author=賈斯柏‧貝克 
| translator=姜和平  
| title=《餓鬼：毛時代大饑荒揭秘》
| publisher=明鏡出版社 
| date=2005年10月 
| url=http://books.google.com/books?id=hjpdAAAAIAAJ 
| isbn=978-1-932138-30-6
| ref = {{SfnRef|貝克|2005}}}}
* [[:w:有線電視|有線電視]]財經資訊台《神州穿梭》 「文革廣西武宣縣紅衛兵吃人肉事件」
{{refend}}

== 外部链接 ==
*[[:w:钱理群|钱理群]]：《[http://www.aisixiang.com/data/3951-2.html 钱理群：说“食人”——周氏兄弟改造国民性思想之一]》{{Wayback|url=http://web.archive.org/web/20150605170543/http://www.aisixiang.com/data/3951-2.html |date=20150605170543 }}

[[Category:History of China]]</text>
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Among all major civilizations worldwide, China has the most recorded instances of cannibalism.&lt;ref&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社, 1994, &quot;中国封建时代的有关（食人习俗的）文字记载是极为丰富的。可以说，中国封建时代的食人习俗证据远比其他时代或其他国家为多&quot;&lt;/ref&gt; This entry documented 388 cannibalism cases recorded in 530 instances from the ''Twenty-Five Histories'' ([[w:Twenty-Four Histories|Twenty-Four Histories]] and [[w:Draft History of Qing|Draft History of Qing]]), consistent with prior research &lt;ref name=鄭麒來統計&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社，1994年版，第153-154页。&lt;/ref&gt;. According to another study, the [[w:Complete Classics Collection of Ancient China|Complete Classics Collection of Ancient China]], a comprehensive Chinese encyclopedic work, recorded 653 cases of filial piety act involving cutting own flesh to cure parents' illness&lt;ref name=鄭麒來統計/&gt;.

Several factors are generally considered responsible for this prevalence. 

* China experienced more famines than any other major civilizations.&lt;ref&gt;邓拓，《中国救荒史》，1937年，“我国灾荒之多，世界罕有，就文献可考的记载来看，从公元前十八世纪，直到公元二十世纪的今日，将近四千年间，几于无年无灾，也几乎无年无荒。西欧学者甚至称我国为‘饥荒的国度’（The Land of Famine)。” &lt;/ref&gt; 
* China experienced the most frequent and intense conflicts among major civilizations.&lt;ref&gt;秦晖，《中国历史上，何来如此深仇大恨》，“中国秦以后历代王朝的寿命不但比‘封建’时代的周‘王朝’和欧洲、日本的宗主王系（不是dynasty）短很多，其‘改朝换代’的巨大破坏性更几乎是人类历史上独有的。……世界史上别的民族有遭到外来者屠杀而种族灭绝的，有毁灭于庞贝式的自然灾变的，但像中国这样残忍的自相残杀确实难找他例。”&lt;/ref&gt; &lt;ref&gt; 福山《政治秩序的起源》，2014年，广西师范大学出版社，第7章，“与其他军事化社会相比，周朝的中国异常残暴。有个估计，秦国成功动员了其总人口的8%到20%，而古罗马共和国的仅1%，希腊提洛同盟的仅5.2%，欧洲早期现代则更低”&lt;/ref&gt; 
* Specific cultural beliefs developed in China, including:
** Rationalizing cannibalism as a means of expressing animosity&lt;ref&gt;《左传·襄公二十一年》，“然二子者，譬如禽兽，臣食其肉而寝处其皮矣”；岳飞，《满江红》，“壮志饥餐胡虏肉，笑谈渴饮匈奴血”；《三国演义》、《水浒传》多处有吃仇人肉的描写；等等&lt;/ref&gt;.
** Attributing medicinal properties to human flesh &lt;ref&gt;唐，陈藏器，《本草拾遗》；明，李时珍，《本草纲目》&lt;/ref&gt;.
** Viewing the practice of cutting own flesh to treat elder relatives as a noble demonstration of filial piety&lt;ref&gt; 《宋史· 卷四百五十六·列传第二百一十五·孝义》：“太祖、太宗以来，……刲股割肝，咸见褒赏；”&lt;/ref&gt;
* China established a comprehensive official historical record system early on, which remained functional even during periods of significant social chaos, preserving extensive historical documentation.

==Statistics==

Key-Ray Chong categorized records of cannibalism within the Twenty-Five Histories, based on their causes.&lt;ref name=&quot;鄭麒來統計&quot; /&gt;

{| class=&quot;wikitable&quot;
|-
!Historical Records!!Subtotal!!Wartime Famine!!Wartime Hatred!!Natural Disasters!!Peace-time Hatred!!Loyalty!!Filial Piety!!Taste!!Other
|-
| [[:w:Shiji|Records of the Grand Historian(''Shiji'')]]||19||6||11	|| ||2|| || || ||
|-
| [[:w:Book of Han|Book of Han]] ||25||11||1||13|| || || || ||
|-
| [[:w:Book of the Later Han|Book of the Later Han]]||26||15|| ||11 ||||||||||
|-
| [[:w:Records of the Three Kingdoms|Records of the Three Kingdoms]]||7||4|| ||3 ||||||||||
|-
| [[:w:Book of Jin|Book of Jin]]||32||16||1||13||2 ||||||||
|-
| [[:w:Book of Wei|Book of Wei]]||8||6||1||1	||||||||||
|-
| [[:w:History of the Southern Dynasties|History of the Southern Dynasties]]||18||12||3||3 ||||||||||
|-
| [[:w:History of the Northern Dynasties|History of the Northern Dynasties]]||6||3||3 ||||||||||||
|-
| [[:w:Book of Northern Qi|Book of Northern Qi]]||2||2 ||||||||||||||
|-
| [[:w:Book of Song|Book of Song]]||2||1||1 ||||||||||||
|-
| [[:w:Book of Liang|Book of Liang]]||9||5||2||2 ||||||||||
|-
| [[:w:Book of Chen|Book of Chen]]||1||1 ||||||||||||||
|-
| [[:w:Book of Sui|Book of Sui]]||8||2||3||3||||||||||
|-
| [[:w:Historical Records of the Five Dynasties|Historical Records of the Five Dynasties]]||15||10||4|| || || ||1||||
|-
| [[:w:Old History of the Five Dynasties|Old History of the Five Dynasties]]||5||3||1||1||||||||||
|-
| [[:w:History of Jin|History of Jin]]||3||||||3||||||||||
|-
| [[:w:History of Liao|History of Liao]]||1||||||1||||||||||
|-
| [[:w:History of Yuan|History of Yuan]]||46||5||1||27||||||13||||
|-
| [[:w:History of Song (book)|History of Song]]||43||4||4||14||||||20||1	||
|-
| [[:w:History of Ming|History of Ming]]||45||5||||22||||||17	||1||
|-
| [[w:Draft History of Qing|Draft History of Qing]]||76||3||||15	||||||58||||
|-
!Total!!397!!114!!36!!132!!4!!0!!109!!2	!!

|}

However, this statistics is incomplete and partially incorrect. It omitted [[:w:Book of Zhou|Book of Zhou]], [[:w:Book of Qi|Book of Southern Qi]], [[:w:Old Book of Tang|Old Book of Tang]], [[:w:New Book of Tang|New Book of Tang]] originally included in the ''Twenty-Five Histories,'' and failed to remove duplicated records in [[:w:History of Ming|History of Ming]].

In addition to previous research, Key-Ray Chong compiled 653 cases of filial piety act involving cutting one's own flesh to cure relatives in [[w:Complete Classics Collection of Ancient China|Complete Classics Collection of Ancient China]], of which 99% involved women, and 56% of these cases involved daughters-in-law cutting their own flesh for their mothers-in-law. Although this polarization may be the result of intentional selection bias, as both male and female cases of flesh-cutting to cure relatives are well documented in the ''Twenty-Five Histories.'' 

Key-Ray Chong concluded:&lt;ref&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社，1994年版，第5-8页。&lt;/ref&gt;

{{Blockquote|text=Chinese practice of survival cannibalism does not significantly differ from that of other cultures; However, &quot;learned cannibalism''(習得性食人)''&quot; in China earned unique characteristics, particularly in its historical prevalence and specific motivations.

Unlike many other regions, where religion played a central role in cannibalistic rituals, Chinese practices were largely secular, often driven by two emotional extremes: '''Virtue and Affection''', including acts performed out of loyalty (尽忠), filial piety (尽孝), or deep love. '''Vengeance and Hatred''', on the other hand, are acts performed for revenge (報仇), to wash away shames (雪恥), or out of pure animosity. To give an example, During wartimes, cannibalism was frequently practiced as a symbolic and literal act of consuming the enemy, rooted in deep-seated hatred.

It is worth noting that ''learned cannibalism'' was also associated with '''culinary appreciation''' or '''medicinal therapy''' among the upper classes. Human flesh was perceived as both a food source and a potent medicine, especially valued for enhancing sexual function. For example, Li Shizhen's [[:w:Compendium of Materia Medica|Compendium of Materia Medica]] listed 35 human organs or substances used for medicinal purposes.}}
==Xia, Zhou and Shang Dynasty==
Note that early Chinese history often blends myth with oral tradition. While these records lack contemporary archeological evidence, they are also historically significant as they reflect how later generations conceptualized the origins of social norms including cannibalism.

# c. 1940 BCE, Xia Dynasty
#: '''English:''' He [Houyi] relied on his archery and neglected civil affairs... The family retainers killed and boiled him, and fed him to his sons. His sons could not bear to eat him and died at city gate.
#: '''Original:''' {{lang|zh-cn|「……（后羿）恃其射也，不修民事而淫於原獸，棄武羅、伯因、熊髡、圉而用寒浞。……羿猶不悛，將歸自田，家眾殺而亨之。以食其子；其子不忍食諸，死於窮門。」}}
#: '''Source:''' ''Zuo Zhuan'', Chapter of Duke Xiang (《左傳·襄公》)
#  Reign of [[:w:King Weng of Zhou|King Weng of Zhou]], c.1112-1050 BCE 
#: '''English:'''　According to ''Diwang Shiji''(The Century of Emperors), [King] Zhou imprisoned King Wen(of Zhou Dynasty). King Wen's eldest son, Boyi Kao, was serving as a hostage in Yin and acted as a charioteer for King Zhou. King Zhou boiled [Boyi Kao] to make a meat soup and presented it to King Wen, saying: &quot;''A true sage should not eat a soup made of his own son.''&quot; 
#: King Wen ate it. King Zhou then remarked, &quot;Who was it said the Earl of the West (King Wen) was a sage? He ate a soup made of his own son without even realizing it.&quot;
#: '''Original:''' 「《帝王世紀》云，（紂）囚文王，文王之長子曰伯邑考，質於殷，為紂御。紂烹為羮，賜文王曰：聖人當不食其子羮。文王食之，紂曰，誰謂西伯聖者，食其子羮尚不知也。」
#: '''Source:''' Justice in History, book 3, records of Yin (《史記正義·卷三·殷本紀》)
#: '''Note:''' The ''Century of Emperors''(《帝王世紀》) cited above was written in [[:w:Jing Dynasty|Jin Period]], and the original is now lost.

== Spring and Autumn / Warring States Periods ==

The [[:w:Spring and Autumn period|Spring and Autumn]] and [[:w:Warring States period|Warring States]] periods (approx. 770–221 BC) marked a significant era where cannibalism was documented under various social and political motivations. Famous Chinese idioms such as &quot;exchanging children to eat&quot; (''易子而食'', from [[:w:Zuo Zhuan|Zuo Zhuan]]) and &quot;eating the flesh and sleeping on the skin&quot; (''食肉寝皮'', from [[:w:Zuo Zhuan|Zuo Zhuan]]) both originated during this time.

Cases of cannibalism during this period can be categorized into four dominant motivations.

# '''Warfare and Siege Famines:''' The most frequent cause. During prolonged sieges, resources were so depleted that citizens resorted to &quot;exchanging children to eat&quot; to avoid consuming their own offspring.
# '''Political motivation:''' A famous case is Yi Ya (易牙), who steamed his own son to serve as a delicacy to Duke Huan of Qi to prove his absolute loyalty.
# '''Intimidation:''' Cannibalism was used as a tool of terror or vengeance. Examples include the Di people killing and eating Duke Yi of Wei(''狄人殺食衛懿公''), or the Ruler of Zhongshan boiling the son of the his own general, Yue Yang(''中山君烹樂羊子''), to test his loyalty.
# '''Cultural customs:''' Early records mention peripheral groups, such as the &quot;People-Eating Kingdom&quot; (啖人國), though these may be the result of Han-centric view of &quot;barbaric&quot; outsiders.

While the [[:w:Zuo Zhuan|Zuo Zhuan]] records at least 15 major natural famines, there are no explicit records of cannibalism resulting from &quot;natural&quot; disasters during this specific period. 

However, historians often note that the absence of such records does not necessarily prove the absence of the practice; rather, it may reflect the selective focus bias on military and political events over lower-class sufferings.

=== Before Warring State period ===

# The practice of &quot;Yi Di&quot; (''宜弟'')
#: '''English''': In the ancient past, there was a kingdom called Kaishu to the east of Yue. When a first-born son was born, they would dismember and eat him. The practice is called &quot;Yi Di&quot; (meaning &quot;benefiting the younger brothers&quot;).
#: '''Original:''' 昔者越之東有輆沭之國者，其長子生，則解而食之，謂之「宜弟」。
#: '''Source:''' ''[[:w:Mozi|Mozi(Book)]]'', &quot;Moderation in Funerals&quot; (《墨子·節葬下》)
# Critique of &quot;Yi Di&quot;, by Mozi
#: '''English:''' Luyang Wenjun said to Mozi: &quot;South of Chu, there is a kingdom of man-eaters called Qiao. When a first-born son is born, they butcher and eat him, calling it 'Yi Di.' If the meat is flavorful, they present it to their ruler, who rewards the father. Is this not a detestable custom?&quot; 
#: Mozi replied: &quot;Even the customs of the Central Kingdoms are similar. How is killing a father and rewarding his son any different from eating a son and rewarding his father? If we do not govern by Benevolence and Righteousness, how can we criticize the barbarians for eating their sons?&quot;
#: '''Original:''' {{lang|zh-tw|魯陽文君語子墨子曰：「楚之南有啖人之國者橋，其國之長子生，則鮮而食之，謂之宜弟。美，則以遺其君，君喜則賞其父。豈不惡俗哉？」子墨子曰：「雖中國之俗，亦猶是也。殺其父而賞其子，何以異食其子而賞其父者哉？苟不用仁義，何以非夷人食其子也？」}}
#: '''Source:''' ''[[:w:Mozi|Mozi(Book)]]'', &quot;Lu Wen&quot; (《墨子·魯問》)
# Ethnographic Records of the Wuhu
#: '''English:''' To the west of the Nanman (Southern Barbarians) lies the Kingdom of Man-eaters, named [[:w:Cochin|Cochin]](Crossed rivers). There, man and woman bath in the same river, thus the name.
#: It is their custom to always dismember and eat the first-born son, calling it &quot;Yi Di.&quot; If the taste is delicious, they offer it to their ruler, who in turn rewards the father. Furthermore, if a man marries a beautiful wife, he offer her to his elder brother. These people are known today as the Wuhu.
#: '''Original:''' {{lang|zh-tw|其西有啖人國，生首子輒解而食之，謂之宜弟。味旨，則以遺其君，君喜而賞其父。取妻美，則讓其兄。今烏滸人是也。}}
#: '''Source:''' ''[[:w:Book of the Later Han|Book of the Later Han]]'', &quot;On the Southern and Southwestern Barbarians&quot; (《後漢書·南蠻西南夷列傳》)

=== In Warring State period ===
# During reigns of Duke Huan of Qi (''齊桓公'', r. 685–643 BCE)
#: '''English''': During the reign of Duke Huan of Qi, Yi Ya served the Duke as his personal chef. The Duke once said that he had never tasted steamed infant. Upon hearing this, Yi Ya steamed his own firstborn son and presented the dish to the Duke. Human nature is such that one loves one's own children; yet he who does not love his own son. Then, what he would do to his own lord?
#: '''Original:''' 夫易牙以调和事（齐桓）公，公曰&quot;惟蒸婴儿之未尝&quot;，于是蒸其首子而献之公。人情非不爱其子也，于子之不爱，将何有于公？
#: '''Source:''' ''[[:w:Guanzi (text)|Guanzi]]'', &quot;Minor Exaltation&quot; (《管子·小称》)
## Alternate records of &quot;Yi Ya&quot;, During reigns of Duke Huan of Qi (''齊桓公'', r. 685–643 BCE)
##: '''English''': Duke Huan of Qi was fond of rare delicacies, and so Yi Ya steamed his own son's head and presented it to him.
##: '''Original:''' 齐桓公好味，易牙蒸其子首而进之。
##: '''Source:''' ''[[:w:Han Feizi|Han Feizi]]'', &quot;The Two Handles&quot; (《韓非子·二柄·難一》)
# 660 BCE: The Death and Consumption of Duke Yi of Wei (''衛懿公'')
#: '''English''': The Di people arrived and overtook Duke Yi of Wei at Rongze, where they killed him. They consumed all of his flesh, only his liver was untouched.
#: '''Original:''' 狄人至，及（卫）懿公于荣泽，杀之，尽食其肉，独舍其肝。
#: '''Source:''' ''[[:w:Lüshi Chunqiu|Lüshi Chunqiu]]'' (《吕氏春秋》)
# 594 BCE: The Siege of Song
#: '''English''': The people of Song, fearing for their lives, sent Hua Yuan on a secret night mission into the Chu encampment. He climbed into the bed of Zi Fan and roused him, saying: &quot;Our lord has sent me, Yuan, to convey our dire situation: our city is reduced to trading children for food and splitting bones for fuel. Even so, a covenant made beneath the city walls — one that would mean the ruin of our state — we cannot accept. Withdraw thirty li (''unit of length, approx. 3 kilometers long)'' from us, and we will obey every command.&quot;
#: '''Original:''' 宋人惧，使华元夜入楚师，登子反之床，起之曰：&quot;寡君使元以病告，曰：'敝邑易子而食，析骸以爨。虽然，城下之盟，有以国毙，不能从也。去我三十里，唯命是听。'&quot;
#: '''Source:''' ''[[:w:Zuo Zhuan|Zuo Zhuan]]'', &quot;The Fifteenth Year of Duke Xuan&quot; (《左傳·宣公十五年》)
## 594 BCE: The Siege of Song  (alternate account)
##: '''English''': In the twentieth year of his reign, King Zhuang of Chu besieged Song in retaliation for the killing of a Chu envoy. After a siege of five months, the food supply within the city was completely exhausted. The inhabitants resorted to trading children for food and burning bones for fuel. Hua Yuan of Song went out to truthfully convey the situation to King Zhuang. The King said: &quot;Truly a man of virtue!&quot; and thereupon withdrew his forces.
##: '''Original:''' 二十年，（楚）围宋，以杀楚使也。围宋五月，城中食尽，易子而食，析骨而炊。宋华元出告以情。庄王曰：&quot;君子哉！&quot;遂罢兵去。
##: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Hereditary Houses of Chu, Vol. 40&quot; (《史記·卷四十·楚世家第十》)
# c. 500 BCE: Zhi the Robber (''盜跖'')
#: '''English''': Confucius and Liuxia Ji were friends; Liuxia Ji's younger brother was named Zhi the Robber. Zhi the Robber commanded a following of nine thousand men, swept through the empire with impunity, plundering the various lords. 
#: He stormed into dwellings, stole cattle and horses, and abducted women. Driven by greed, he cast aside all bonds of kinship, disregarding his parents and siblings, and made no offerings to his ancestors.
#: Wherever his forces passed, large states fortified their walls and small states withdrew into strongholds, and all the people suffered greatly. [...] At that time, Zhi the Robber was resting his men on the southern slope of Mount Tai, mincing human livers and eating them.
#: '''Original:''' 孔子与柳下季为友，柳下季之弟名曰盗跖。盗跖从卒九千人，横行天下，侵暴诸侯；穴室枢户，驱人牛马，取人妇女；贪得忘亲，不顾父母兄弟，不祭先祖。所过之邑，大国守城，小国入保，万民苦之。……盗跖乃方休卒徒太山之阳，脍人肝而餔之。
#: '''Source:''' ''[[:w:Zhuangzi (book)|Zhuangzi]]'', &quot;Robber Zhi&quot; (《莊子·盜跖》)
# 409 BCE: Yue Yang Drinks His Son's Broth
#: '''English''': Yue Yang served as a general of Wei and led an attack on Zhongshan. His son was residing in Zhongshan at the time, and the ruler of Zhongshan had the son boiled and sent the resulting broth to Yue Yang. Yue Yang sat beneath his campaign tent and drank it, finishing the entire cup. 
#: Marquis Wen of Wei said to his advisor Du Shize: &quot;Yue Yang, for my sake, ate the flesh of his own son.&quot; Du replied: &quot;One who can eat his own son's flesh. Who would he not eat?&quot; After Yue Yang had pacified Zhongshan, Marquis Wen rewarded his achievement but harbored doubts about his character.
#: '''Original:''' 乐羊为魏将而攻中山。其子在中山，中山之君烹其子而遗之羹，乐羊坐于幕下而啜之，尽一杯。文侯谓睹师赞曰：&quot;乐羊以我之故，食其子之肉。&quot;赞对曰：&quot;其子之肉尚食之，其谁不食？&quot;乐羊既罢中山，文侯赏其功而疑其心。
#: '''Source:''' ''[[:w:Zhanguo Ce|Zhanguo Ce]]'', &quot;Stratagems of Wei I, Vol. 22&quot; (《戰國策·卷二十二·魏策一》)
# 403 BCE: The Siege of Jinyang ''(晉陽之戰'')
#: '''English''': The three states of Zhi, Wei, and Han besieged Jinyang for over a year, and then diverted the Fen River to flood the city. The floodwaters rose to within three planks' breadth of the top of the walls. Within the city, cauldrons were suspended over fires for cooking, inhabitants exchanged children to eat.
#: '''Original:''' 三国（智魏韩）攻晋阳，岁馀，引汾水灌其城，城不浸者三版。城中悬釜而炊，易子而食。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Clan of Zhao, Vol. 43&quot; (《史記·卷四十三·趙世家第十三》)
## 403 BCE: The Siege of Jinyang (alternate record)
##: '''English''': The three clans of Zhi, Wei, and Han encircled the people of Zhao at Jinyang and flooded the city; the floodwaters rose to within three planks' breadth of the top of the walls, and the inhabitants resorted to eating men and horses.
##: '''Original:''' 三家（智魏韩）以国人围（赵国晋阳）而灌之，城不浸者三版，人马相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 1 (《資治通鑑·卷一》)
# 260 BCE: The Battle of Changping (''長平之戰'')
#: '''English''': By the ninth month, the Zhao soldiers had been without food for forty-six days, and in secret they began killing and ate each other.
#: '''Original:''' 至九月，赵卒不得食四十六日，皆内阴相杀食。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Bai Qi and Wang Jian, Vol. 73&quot; (《史記·卷七十三·白起王翦列傳第十三》)
## 260 BCE: The Battle of Changping (alternate record)
##: '''English''': The Zhao army was cut off from food for forty-six days, during which they secretly killed and ate each other.
##: '''Original:''' 赵军食绝四十六日，皆内阴相杀食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 5 (《資治通鑑·卷五》)
# 257 BCE: Li Tong(''李同'')'s Appeal at the Siege of Handan
#: '''English''': Li Tong said: &quot;The people of Handan are burning bones for fuel and trading children for food. Their plight could not be more desperate. Yet in your household, hundreds of concubines and maids are clothed in fine silk, with surplus grain and meat to spare, while the common people cannot complete a garment of coarse cloth and cannot fill themselves even with dregs and husks.&quot;
#: '''Original:''' 邯郸之民，炊骨易子而食，可谓急矣，而君之後宫以百数，婢妾被绮縠，馀粱肉，而民褐衣不完，糟糠不厌。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Lord Pingyuan and Yu Qing, Vol. 76&quot; (《史記·卷七十六·平原君虞卿列傳第十六》)
# c. 250 BCE: The Siege of Liaocheng
#: '''English''': Qi's general Tian Dan besieged Liaocheng for over a year, with heavy casualties among his troops, yet the city did not fall. Lu Zhonglian then composed a letter, tied it to an arrow, and shot it into the city, addressed to the Yan commander. The letter read: &quot;[...] Now you hold the exhausted people of Liaocheng against the full force of Qi's army — this is the defensive resolve of Mozi. Your men eat others and burn their bones for fuel, yet none harbor thoughts of surrender — this is the military discipline of Sun Bin. Your name shall be known throughout the realm.&quot;
#: '''Original:''' 齐田单攻聊城岁馀，士卒多死而聊城不下。鲁连乃为书，约之矢以射城中，遗燕将。书曰：……今公又以敝聊之民距全齐之兵，是墨翟之守也。食人炊骨，士无反外之心，是孙膑之兵也。能见於天下。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Lu Zhonglian and Zou Yang, Vol. 83&quot; (《史記·卷八十三·魯仲連鄒陽列傳第二十三》)

==Han Dynasty==
The wars between the Qin and Han dynasties caused large-scale famine and population decline across China, a pattern that would recur with nearly every subsequent dynastic transition.

# Early Han Dynasty: Famine and Cannibalism Following the Collapse of Qin
#: '''English''': At the founding of the Han dynasty, inheriting the devastation left by Qin, the various lords rose simultaneously in conflict. The people abandoned their livelihoods, and a great famine ensued. Price of one shi of rice reached five thousand coins; people ate each other, more than half the population perished. Emperor Gaozu then issued an order permitting the people to sell their children, and directed the starving to seek food in Shu and Han.
#: '''Original:''' 汉兴，接秦之敝，诸侯并起，民失作业而大饥馑。凡米石五千，人相食，死者过半。高祖乃令民得卖子，就食蜀、汉。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 205 BCE: Great Famine in Guanzhong, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': A great famine struck Guanzhong; the price of one hu of rice reached ten thousand coins, and people ate each other. The people were directed to seek food in Shu and Han.
#: '''Original:''' 关中大饥，米斛万钱，人相食。令民就食蜀、汉。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Gao, Vol. 1a&quot; (《漢書·卷一上·高帝紀第一上》)
## 205 BCE: Great Famine in Guanzhong, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': A great famine struck Guanzhong; the price of one hu of rice reached ten thousand coins, and people ate each other.
##: '''Original:''' 关中大饥，米斛万钱，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 9 (《資治通鑑·卷九》)
# 196 BCE: Minced flesh of Peng Yue, in ''[[:w:Records of the Grand Historian|Shiji]]''
#: '''English''': In the eleventh year, Empress Gao put to death the Marquis of Huaiyin; (Ying) Bu grew fearful at heart. In summer, Han executed Liang Wang Peng Yue, minced his flesh into paste, and sent portions of his flesh to all the lords. 
#:When it reached Huainan, the King of Huainan was out hunting; upon beholding the paste, he trembled greatly, and secretly ordered men to muster troops, watching for signs of trouble in the neighboring commanderies.
#: '''Original:''' 十一年，高后诛淮阴侯，布因心恐。夏，汉诛梁王彭越，醢之，盛其醢遍赐诸侯。至淮南，淮南王方猎，见醢，因大恐，阴令人部聚兵，候伺旁郡警急。
#: '''Source:''' ''[[:w:Records of the Grand Historian|Shiji]]'', &quot;Biography of Qing Bu&quot; (《史记·卷九十一·黥布列传第十三》)
# 138 BCE: Flood and Famine on the Yellow River Plain, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the third spring of that year, the Yellow River overflowed onto the Pingyuan plain. Great famine, people ate each other.
#: '''Original:''' 三年春，河水溢于平原，大饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Wu, Vol. 6&quot; (《漢書·卷六·武帝紀第六》)
## 138 BCE: Flood and Famine on the Yellow River Plain, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': The Yellow River overflowed onto the Pingyuan plain. Great famine, people ate each other.
##: '''Original:''' 河水溢于平原。大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 17 (《資治通鑑·卷十七》)
# 135 BCE: Ji An's Report on Famine in Henei, ''[[:w:Shiji|Shiji]]''
#: '''English''': Ji An returned and reported: &quot;A household fire has spread to neighboring houses. it is not worth undue concern. On my way, I passed through Henan, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another. I therefore took it upon myself, acting on temporary authority, to use the imperial tally to open the Henan granaries and relieve the destitute people. I now request to return the tally and submit to punishment for acting beyond my authority.&quot; The Emperor, recognizing his virtue, pardoned him.
#: '''Original:''' 还报曰：&quot;家人失火，屋比延烧，不足忧也。臣过河南，河南贫人伤水旱万余家，或父子相食，臣谨以便宜，持节发河南仓粟以振贫民。臣请归节，伏矫制之罪。&quot;上贤而释之。
#: '''Source:''' ''[[:w:Shiji|Shiji]]'', &quot;Biographies of Ji An and Zheng Dangshi, Vol. 120&quot; (《史記·卷一百二十·汲鄭列傳第六十》)
## 135 BCE: Ji An's Report on Famine in Henei, ''[[:w:Book of Han|Book of Han]]''
##: '''English''': [Ji An] returned and reported: &quot;A household fire has spread to neighboring houses — it is not worth undue concern. On my way, I passed through Henei, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another. I therefore took it upon myself, acting on temporary authority, to use the imperial tally to open the Henei granaries and relieve the destitute people. I request to return the tally and submit to punishment for acting beyond my authority.&quot; The Emperor, recognizing his virtue, pardoned him and transferred him to serve as Prefect of Xingyang.
##: '''Original:''' 还报曰：&quot;家人失火，屋比延烧，不足忧。臣过河内，河内贫人伤水旱万余家，或父子相食，臣谨以便宜，持节发河内仓粟以振贫民。请归节，伏矫制罚。&quot;上贤而释之，迁为荥阳令。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Zhang, Feng, Ji, and Zheng, Vol. 50&quot; (《漢書·卷五十·張馮汲鄭傳第二十》)
## 135 BCE: Ji An's Report, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': On my way, I passed through Henan, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another.
##: '''Original:''' 臣过河南，河南贫人伤水旱万馀家，或父子相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 17 (《資治通鑑·卷十七》)
# 114 BCE: Famine in Shandong, ''[[:w:Shiji|Shiji]]''
#: '''English''': At that time, the eastern provinces had suffered from Yellow River floods, and for several consecutive years the harvests had failed. In some places, spanning one to two thousand li, people resorted to eating one another.
#: '''Original:''' 是时山东被河灾，及岁不登数年，人或相食，方一二千里。
#: '''Source:''' ''[[:w:Shiji|Shiji]]'', &quot;Treatise on Equalization, Vol. 30&quot; (《史記·卷三十·平準書第八》)
## 114 BCE: Famine in Shandong(the East), ''[[:w:Book of Han|Book of Han]](1)''
##: '''English''': At that time, the eastern provinces had suffered from Yellow River floods, and for several consecutive years the harvests had failed. In some places, spanning two to three thousand li, people resorted to eating one another. The Emperor, moved by compassion, ordered the famine victims to travel and seek food in the Yangtze and Huai River regions, and those who wished to remain were permitted to settle there. Imperial envoys with carriages and canopies followed one another on the roads to escort them, and grain from Ba and Shu was dispatched to provide relief.
##: '''Original:''' 是时山东被河灾，乃岁不登数年，人或相食，方二三千里。天子怜之，令饥民得流就食江、淮间，欲留，留处。使者冠盖相属于道护之，下巴、蜀粟以赈焉。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24b&quot; (《漢書·卷二十四下·食貨志第四下》)
## 114 BCE: Famine in the East, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In the third month of the third Yuanding year, water froze; in the fourth month, snow fell. In more than ten commanderies east of the passes, people ate each other.
##: '''Original:''' 元鼎三年三月水冰，四月雨雪，关东十余郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on the Five Elements, Vol. 27&quot; (《漢書·卷二十七中之下·五行志第七中之下》)
## 114 BCE: Famine in the East, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': More than forty commanderies and kingdoms east of the passes suffered famine, people ate each other.
##: '''Original:''' 关东郡、国四十馀饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 20 (《資治通鑑·卷二十》)
# 113 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In summer, the fourth month, hail fell. In more than ten commanderies and kingdoms east of the passes, Great Famine; people ate each other.
#: '''Original:''' 夏四月，雨雹，关东郡国十余饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Wu, Vol. 6&quot; (《漢書·卷六·武帝紀第六》)
# 141–87 BCE: Critique of Emperor Wu's Reign, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': &quot;Though Emperor Wu had merit in driving back the four barbarians and expanding the realm, yet he slew great numbers of his men, exhausted the people's wealth, indulged in extravagance without measure. 
#: The realm was left hollow and depleted, the hundred folk scattered and adrift, half perished. Locusts rose in great swarms, scorching the earth for thousands of li; in some places people ate each other, and the stores have not recovered to this day. 
#: He bestowed no virtue nor grace upon the people, and ought not to have temple rites established in his honour.&quot;
#: '''Original:''' 武帝虽有攘四夷广土斥境之功，然多杀士众，竭民财力，奢泰亡度，天下虚耗，百姓流离，物故者半。蝗虫大起，赤地数千里，或人民相食，畜积至今未复。亡德泽于民，不宜为立庙乐。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Sui, Liang, Xiahou, Jing, Yi and Li, Vol. 75&quot; (《漢書·卷七十五·眭兩夏侯京翼李傳第四十五》)
# c. 104 BCE: Depletion of the Realm After Dong Zhongshu, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': After Zhongshu's death, expenditures grew ever greater, the realm was hollow and depleted, and once more people ate each other.
#: '''Original:''' 仲舒死后，功费愈甚，天下虚耗，人复相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In the ninth month, eleven commanderies and kingdoms east of the passes suffered great floods. Famine; in some places people ate each other. Neighboring commanderies were called upon to render aid in coin and grain.
#: '''Original:''' 九月，关东郡国十一大水，饥，或人相食，转旁郡钱、谷''(穀)''以相救。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Yuan, Vol. 9&quot; (《漢書·卷九·元帝紀第九》)
## 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In the first year of Chuyuan under Emperor Yuan, [...] in the fifth month the Bohai Sea overflowed greatly. In the sixth month, Great Famine struck the east; many among the people starved to death, and in Langye Commandery people ate each other.
##: '''Original:''' 元帝初元元年，……其五月，勃海水大溢。六月，关东大饥，民多饿死，琅邪郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Astronomy, Vol. 26&quot; (《漢書·卷二十六·天文志第六》)
## 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': In autumn, the ninth month, eleven commanderies and kingdoms east of the passes suffered great floods and famine; in some places people ate each other.
##: '''Original:''' 秋，九月，关东郡、国十一大水，饥，或人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 28 (《資治通鑑·卷二十八》)
# 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In the sixth month, famine struck the east; in the land of Qi, people ate each other.
#: '''Original:''' 六月，关东饥，齐地人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Yuan, Vol. 9&quot; (《漢書·卷九·元帝紀第九》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': When Emperor Yuan ascended the throne, great floods struck the realm; eleven eastern commanderies suffered most grievously. In the second year, famine struck the land of Qi; grain reached three hundred coins per shi, many among the people starved to death, and in Langye Commandery people ate each other.
##: '''Original:''' 元帝即位，天下大水，关东郡十一尤甚。二年，齐地饥，谷''(穀)''石三百余，民多饿死，琅邪郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](3)''
##: '''English''': The following year, in the second month, on the day wuwu, the earth shook. That summer, in the land of Liu, people ate each other. [...] Yi Feng memorialized: &quot;The eastern lands have suffered famine for years running, compounded by pestilence; the hundred folk are wan with hunger, and some have come to eat each other. The earth trembles repeatedly, the heavens are turbid, and the light of the sun grows dim.&quot;
##: '''Original:''' 明年二月戊午，地震。其夏，刘地人相食。……（翼奉）上疏曰：……今东方连年饥馑，加之以疾疫，百姓菜色，或至相食。地比震动，天气溷浊，日光侵夺。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Sui, Liang, Xiahou, Jing, Yi and Li, Vol. 75&quot; (《漢書·卷七十五·眭兩夏侯京翼李傳第四十五》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](4)''
##: '''English''': When Emperor Yuan first ascended the throne, he summoned Yu to serve as Remonstrant Counsellor and repeatedly sought his counsel on affairs of governance. At that time the harvests had failed and many commanderies were in distress. 
##: Yu exclaimed: &quot;Now the people die of Great Famine; the dead go unburied and are eaten by dogs and swine. People eat each other, whilst the horses in the imperial stables feed on grain and grow so fat and vigorous that they must be walked daily to work it off. Is this what it means for a sovereign, having received the Mandate of Heaven, to be father and mother to the people?&quot;
##: '''Original:''' 元帝初即位，征禹為諫大夫，數虛己問以政事。是時，年歲不登，郡國多困，禹奏言：[……] 今民大飢而死，死又不葬，為犬豬食。人至相食，而廄馬食粟，苦其大肥，氣甚怒至，乃日步作之。王者受命於天，為民父母，固當若此乎！（
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Wang, Gong, Liang Gong and Bao, Vol. 72&quot; (《漢書·卷七十二·王貢兩龔鮑傳第四十二》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](5)''
##: '''English''': Kuang Heng memorialized: &quot;The eastern lands have suffered famine for years running; the hundred folk are in want and distress, and some have come to eat each other. This hath all arisen from levies and taxes being too heavy, the burdens borne by the people being too great, and the officials failing in their duty to settle and succour them.&quot;
##: '''Original:''' 匡）衡上疏曰：……今关东连年饥馑，百姓乏困，或至相食，此皆生于赋敛多，民所共者大，而吏安集之不称之效也。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Kuang, Zhang, Kong and Ma, Vol. 81&quot; (《漢書·卷八十一·匡張孔馬傳第五十一》)
## 47 BCE: Famine in Qi, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Famine struck the east; in the land of Qi, people ate each other.
##: '''Original:''' 关东饥，齐地人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 28 (《資治通鑑·卷二十八》)
# 17 BCE: Emperor Cheng's Edict Dismissing Xue Xuan, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': Emperor Cheng decreed the dismissal of Xue Xuan, saying: &quot;I, being unenlightened, have seen repeated ill omens; the harvests have failed year upon year, the granaries stand empty, the hundred folk suffer Great Famine, wandering and scattered upon the roads. Those who have perished of pestilence number in the tens of thousands; people eat each other, bandits rise on all sides, and the offices of governance lie neglected. This is owing to mine own want of virtue and the failings of mine own ministers.&quot;
#: '''Original:''' 朕既不明，变异数见，岁比不登，仓廪空虚，百姓饥馑，流离道路，疾疫死者以万数，人至相食，盗贼并兴，群职旷废，是朕之不德而股肱不良也。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Xue Xuan and Zhu Bo, Vol. 83&quot; (《漢書·卷八十三·薛宣朱博傳第五十三》)
# 15 BCE: Floods in Liang and Pingyuan, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the second year of Yongshi, the kingdoms of Liang and Pingyuan suffered floods in consecutive years; people ate each other. The regional inspectors, prefects and chancellors were held accountable and dismissed.
#: '''Original:''' 永始二年，梁国、平原郡比年伤水灾，人相食，刺史、守、相坐免。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 14 CE: Great Famine Along the Frontier, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the first year of Tianfeng under Wang Mang, Great Famine struck the borderlands; people ate each other.
#: '''Original:''' 缘边大饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99b&quot; (《漢書·卷九十九中·王莽傳第六十九中》)
## 14 CE: Great Famine Along the Frontier, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Great Famine struck the borderlands; people ate each other.
##: '''Original:''' 缘边大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 37 (《資治通鑑·卷三十七》)
# 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In his final years, bandits rose in great numbers; armies were dispatched to suppress them, and their officers ran amok beyond the passes. In the northern borderlands and in the lands of Qing and Xu, people ate each other; east of Luoyang, grain reached two thousand coins per shi.
#: '''Original:''' 末年，盗贼群起，发军击之，将吏放纵于外。北边及青、徐地人相食，雒阳以东米石二千。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': [...] battle and slaughter, captivity by the four border peoples, criminal penalties, Great Famine, pestilence, and people eating each other had together reduced the households of the realm by half.
##: '''Original:''' 战斗死亡，缘边四夷所系虏，陷罪，饥疫，人相食，及莽未诛，而天下户口减半矣。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24b&quot; (《漢書·卷二十四下·食貨志第四下》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In that month, the Red Eyebrows slew the Grand Preceptor Xi Zhong Jing Shang. East of the passes, people ate each other.
##: '''Original:''' 是月，赤眉杀太师牺仲景尚。关东人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳第六十九下》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': East of the passes, people ate each other.
##: '''Original:''' 关东人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 38 (《資治通鑑·卷三十八》)
# 23 CE: The Fate of Wang Mang's Corpse, ''Book of Han''
#: '''English''': Wang Mang's severed head was sent to Gengshi and hung in the market of Wan. The common folk vied to strike and beat it; some cut out his tongue and ate it.
#: '''Original:''' 传（王）莽首诣更始，县宛市，百姓共提击之，或切食其舌。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳》)
# 23 CE: Siege of Wan — Cen Peng's Surrender, ''Hou Han Shu''
#: '''English''': When Zhen Fu fell and Cen Peng was wounded, he fled back to Wan and held the city together with Yan Shuo. Han forces besieged them for several months; the city's provisions were exhausted and people ate each other. Peng and Shuo thereupon surrendered the city.
#: '''Original:''' 汉兵攻之数月，城中粮尽，人相食，彭乃与说举城降。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Feng, Cen and Jia, Vol. 17&quot; (《後漢書·卷十七·馮岑賈列傳第七》)
# 23 CE: Siege of Wan — Cen Peng's Surrender, ''Zizhi Tongjian''
#: '''English''': [...] Han forces besieged them for several months. People within the city ate each other; they thereupon surrendered.
#: '''Original:''' 汉兵攻之数月，城中人相食，乃举城降。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 39 (《資治通鑑·卷三十九》)
# 24 CE: Li Xiong's Counsel to Gongsun Shu, ''Hou Han Shu''
#: '''English''': [...] &quot;Now the lands east of the mountains suffer Great Famine; the common folk eat each other. Where armies have passed, cities and towns are left as mounds of rubble.&quot;
#: '''Original:''' 今山东饥馑，人庶相食；兵所屠灭，城邑丘墟。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Wei Xiao and Gongsun Shu, Vol. 13&quot; (《後漢書·卷十三·隗囂公孫述列傳第三》)
# 25 CE: The Red Eyebrows Sack Chang'an, ''Book of Han''
#: '''English''': The Red Eyebrows burned the palaces and markets of Chang'an and slew Gengshi. The starving people ate each other; those who perished numbered in the hundreds of thousands. Chang'an was left a wasteland, and none walked its streets.
#: '''Original:''' 赤眉遂烧长安宫室市里，害更始。民饥饿相食，死者数十万，长安为虚，城中无人行。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳第六十九下》)
# 26 CE: Famine in Guanzhong, ''Hou Han Shu(1)''
#: '''English''': Great Famine struck Guanzhong; people ate each other.
#: '''Original:''' 关中饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Guangwu, Vol. 1a&quot; (《後漢書·卷一上·光武帝紀第一上》)
## 26 CE: Famine in Guanzhong, ''Hou Han Shu(2)''
##: '''English''': At that time, the three adjuncts were in great turmoil; people ate each other, the cities and towns were emptied, white bones lay strewn across the fields, and the survivors gathered here and there in fortified encampments, each holding firm.
##: '''Original:''' 时三辅大乱，人相食，城郭皆空，白骨蔽野，遗人往往聚为营保，各坚守不下。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Xuan and Liu Penzi, Vol. 11&quot; (《後漢書·卷十一·劉玄劉盆子列傳第一》)
## 26 CE: Famine in Guanzhong, ''Zizhi Tongjian''
##: '''English''': Great Famine struck the three adjuncts; people ate each other, the cities and towns were emptied, and white bones lay strewn across the fields.
##: '''Original:''' 三辅大饥，人相食，城郭皆空，白骨蔽野。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 40 (《資治通鑑·卷四十》)
# 27 CE: Siege of Ji, Zizhi Tongjian
#: '''English''': Within Zhu Fu's city of Ji, provisions were exhausted; people ate each other.
#: '''Original:''' 浮城中粮尽，人相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 41 (《資治通鑑·卷四十一》)
## 27 CE: Siege of Ji'', Hou Han Shu''
##: '''English''': Within Fu's city, provisions were exhausted; people ate each other.
##: '''Original:''' 浮城中粮尽，人相食。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Zhu, Feng, Yu, Zheng and Zhou, Vol. 33&quot; (《後漢書·卷三十三·朱馮虞鄭周列傳第二十三》)
# 27 CE: Yan Cen's Retreat to Nanyang, ''Hou Han Shu''
#: '''English''': At that time the people suffered Great Famine and ate each other; one jin of gold could be exchanged for but five sheng of beans. The roads were cut off and supplies could not get through; the soldiers subsisted on wild fruit.
#: '''Original:''' 时，百姓饥饿，人相食，黄金一斤易豆五升。道路断隔，委输不至，军士委以果实为粮。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Feng, Cen and Jia, Vol. 17&quot; (《後漢書·卷十七·馮岑賈列傳第七》)
# 109 CE: Great Famine in the Capital, ''Hou Han Shu''
#: '''English''': In the third month, Great Famine struck the capital; people ate each other.
#: '''Original:''' 三月，京师大饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor An, Vol. 5&quot; (《後漢書·卷五·孝安帝紀第五》)
## 109 CE: Great Famine in the Capital, ''Zizhi Tongjian''
##: '''English''': In the third month, Great Famine struck the capital; people ate each other.
##: '''Original:''' 三月，京师大饥，民相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 49 (《資治通鑑·卷四十九》)
# 109 CE: Floods and Famine Across the Realm, ''Hou Han Shu(1)''
#: '''English''': That year, the capital and forty-one commanderies and kingdoms suffered hail. Great Famine struck Bing and Liang; people ate each other.
#: '''Original:''' 是岁，京师及郡国四十一雨水雹。并、凉二州大饥，人相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor An, Vol. 5&quot; (《後漢書·卷五·孝安帝紀第五》)
## 109 CE: Floods and Famine Across the Realm, ''jin Shu''
##: '''English''': In the third year of Yongchu under Emperor An, floods and drought struck the realm; people ate each other.
##: '''Original:''' 安帝永初三年，天下水旱，人民相食。
##: '''Source:''' ''[[:w:Jin Shu|Jin Shu]]'', &quot;Treatise on Food and Commerce, Vol. 26&quot; (《晉書·卷二十六·志第十六·食貨》)
## 109 CE: Floods and Famine Across the Realm, ''Zizhi Tongjian''
##: '''English''': The capital and forty-one commanderies suffered floods; Great Famine struck Bing and Liang; people ate each other.
##: '''Original:''' 京师及郡国四十一雨水，并、凉二州大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 49 (《資治通鑑·卷四十九》)
# 151 CE: Drought and Famine, ''Hou Han Shu''
#: '''English''': Drought struck the capital. Great Famine afflicted Rencheng and Liang; people ate each other.
#: '''Original:''' 京师旱。任城、梁国饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Huan, Vol. 7&quot; (《後漢書·卷七·孝桓帝紀第七》)
## 151 CE: Drought and Famine, ''Zizhi Tongjian''
##: '''English''': Drought struck the capital; Great Famine afflicted Rencheng and Liang; people ate each other.
##: '''Original:''' 京师旱，任城、梁国饥，民相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 53 (《資治通鑑·卷五十三》)
# 155 CE: Famine in Sili and Jizhou, ''Hou Han Shu''
#: '''English''': In the second month, famine struck Sili and Jizhou; people ate each other.
#: '''Original:''' 二月，司隶、冀州饥，人相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Huan, Vol. 7&quot; (《後漢書·卷七·孝桓帝紀第七》)
# 155 CE: Famine in Sili and Jizhou, ''Zizhi Tongjian''
#: '''English''': In the second month, famine struck Sili and Jizhou; people ate each other.
#: '''Original:''' 二月，司隶、冀州饥，人相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 53 (《資治通鑑·卷五十三》)
# 170 CE: Spousal Cannibalism in Henei and Henan, ''Hou Han Shu''
#: '''English''': In the first month of spring in the third year of Jianning, in Henei wives ate their husbands, and in Henan husbands ate their wives.
#: '''Original:''' 三年春正月，河内人妇食夫，河南人夫食妇。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Ling, Vol. 8&quot; (《後漢書·卷八·孝靈帝紀第八》)
# 194 CE: Great Drought in the Three Adjuncts, ''Hou Han Shu''
#: '''English''': A great drought struck the three adjuncts from the fourth month to this day. At that time one hu of grain fetched fifty thousand coins, and one hu of beans or wheat twenty thousand. People ate each other; white bones lay heaped in piles.
#: '''Original:''' 三辅大旱，自四月至于是月。是时谷一斛五十万，豆麦一斛二十万，人相食啖，白骨委积。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Xian, Vol. 9&quot; (《後漢書·卷九·孝獻帝紀第九》)
## 194 CE: Great Drought in the Three Adjuncts, ''Zizhi Tongjian''
##: '''English''': From the fourth month no rain fell. One hu of grain was worth fifty thousand coins; within Chang'an, people ate each other.
##: '''Original:''' 自四月不雨至于是月，谷一斛直钱五十万，长安中人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# Liu Ping Spared by Cannibals, ''Hou Han Shu''
#: '''English''': Liu Ping, styled Gongzi, was a man of Pengcheng in Chu. During the upheavals of the Gengshi era, he and his mother hid together in the wilderness. 
#: One morning he went out to forage for food and was seized by starving bandits who meant to boil and eat him. He knelt and said: &quot;This morning I went to gather herbs for my aged mother, who depends on me for her life. I beg ye to let me return, feed my mother, and then come back to die.&quot; Tears streamed down his face. 
#: The bandits, moved by his sincerity, took pity and released him. Liu Ping returned, fed his mother, and then told her: &quot;I made a pledge to the bandits; honour forbids me to deceive them.&quot; He went back to the bandits. They were all greatly astonished and said to one another: &quot;We have long heard of men of fierce integrity — now we behold one. Go, friend; we have not the heart to eat thee.&quot; And so he was spared.
#: '''Original:''' 刘平字公子，楚郡彭城人也。[…] 更始时，天下乱，[…] 与母俱匿野泽中。平朝出求食，逢饿贼，将亨（通“烹”）之，平叩头曰：“今旦为老母求菜，老母待旷为命，愿得先归，食母毕，还就死。”因涕泣。贼见其至诚，哀而遣之。平还，既食母讫，因白曰：“属与贼期，义不可欺。”遂还诣贼。众皆大惊，相谓曰：“常闻烈士，乃今见之。子去矣，吾不忍食子。”于是得全。（《后汉书·卷三十九·刘赵淳于江刘周赵列传第二十九》）
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Zhao Xiao Offers Himself to Cannibals, ''Hou Han Shu''
#: '''English''': [After the fall of Wang Mang] the realm fell into turmoil and people ate each other. [Zhao Xiao's] younger brother Li was seized by starving bandits. 
#: Upon hearing this, Zhao Xiao bound himself and went to the bandits, saying: &quot;Li hath long been starved and is thin and gaunt; I filleth ye hunger better than him&quot; The bandits were greatly astonished and released them both, saying: &quot;Go home for now, and bring back rice and dried provisions instead.&quot; 
#: Xiao sought provisions but could find none; he returned to the bandits and offered himself for the pot. The bandits, marvelling at him, did him no harm.
#: '''Original:''' （王莽之後）天下乱，人相食。孝弟礼为饿贼所得，孝闻之，即自缚诣贼，曰：&quot;礼久饿羸瘦，不如孝肥饱。&quot;贼大惊，并放之，谓曰：&quot;可且归，更持米糒来。&quot;孝求不能得，复往报贼，愿就亨。众异之，遂不害。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Wang Lin Guards His Parents' Tomb, ''Hou Han Shu''
#: '''English''': In Runan there was a man named Wang Lin, a junior official, who lost his parents when he was but ten years of age. 
#: When great turmoil broke out and the people fled, only Wang Lin and his brothers remained to guard the burial mound, their weeping unceasing. His younger brother Ji went out and was seized by the Red Eyebrows, who meant to eat him. Wang Lin bound himself and begged to die in his brother's stead. 
#: The bandits, moved to pity, released them both; and by this deed Wang Lin's name became renowned throughout his hometown.
#: '''Original:''' 汝南有王琳巨尉者，年十余岁丧父母。因遭大乱，百姓奔逃，惟琳兄弟独守冢庐，号泣不绝。弟季，出遇赤眉，将为所哺，琳自缚，请先季死，贼矜而放遣，由是显名乡邑。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Wei Tan Spares His Fellow Captives, ''Hou Han Shu''
#: '''English''': Wei Tan of Langye, styled Shaoxian, was likewise seized by starved bandits. Several dozen captives were bound and awaited their turn to be boiled. 
#: The bandits, seeing that Tan appeared honest and trustworthy, set him apart to tend the cooking fire, though they bound him again each evening. Among the bandits was one Yi Changgong, who took especial pity on Tan; he secretly loosened Tan's bonds and said: &quot;Ye are all destined to be eaten; flee hence at once.&quot; 
#: Tan replied: &quot;I have tended the fire for ye, there I always had some leavings for myself; the others have been fed only on grass and weeds; better to eat (''relatively well-fed'') me instead.&quot; Changgong, moved by his righteousness, persuaded the others to release all the captives, and all were spared.
#: '''Original:''' 琅邪魏谭少闲者，时亦为饥寇所获，等辈数十人皆束缚，以次当亨（通“烹”）。贼见谭似谨厚，独令主爨，暮辄执缚。贼有夷长公，特哀念谭，密解其缚，语曰：&quot;汝曹皆应就食，急从此去。&quot;对曰：&quot;谭为诸君爨，恒得遗余，余人皆菇草莱，不如食我。&quot;长公义之，相晓赦遣，并得俱免。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Er Meng and Che Cheng Offer Themselves for Each Other, ''Hou Han Shu''
#: '''English''': Er Meng Ziming of Qi and Che Cheng Ziwei of Liangjun, brothers, were seized together by the Red Eyebrows and were about to be eaten. Meng and Cheng knelt and each begged to die in the other's stead. The bandits, moved to pity, released them both.
#: '''Original:''' 齐国兒萌子明、梁郡车成子威二人，兄弟并见执于赤眉，将食之，萌、成叩头，乞以身代，贼亦哀而两释焉。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Chunyu Gong Offers Himself for His Brother, ''Hou Han Shu''
#: '''English''': Chunyu Gong, styled Mengsun, was a man of Chunyu in Beihai. […] At the end of Wang Mang's reign, when famine and war arose, his elder brother Chong was seized by bandits who meant to boil and eat him. Gong begged to take his brother's place; both were released.
#: '''Original:''' 淳于恭字孟孙，北海淳于人也。[…] 王莽末，岁饥兵起，恭兄崇将为盗所亨，恭请代，得俱免。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)

== Three Kingdoms period ==
According to population statics at the time, the population of the Three Kingdoms period was only one-seventh of that during the reign of Emperor Huan of the Eastern Han Dynasty.&lt;ref&gt;秦晖，《中国历史上，何来如此深仇大恨》&lt;/ref&gt; This was the largest population decrease in Chinese history, evidenced by Cao Cao's poem; &quot;Pale bones exposed in wild fields, no crowing of roosters heard throughout thousands of li&quot; (白骨露于野，千里无鸡鸣).
# 194 CE: Famine During the Puyang Campaign, ''Sanguozhi''
#: '''English''': That year, one hu of grain fetched over fifty thousand coins; people ate each other. Newly recruited troops were thereupon disbanded.
#: '''Original:''' 是岁谷一斛五十余万钱，人相食，乃罢吏兵新募者。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Annals of Emperor Wu, Vol. 1&quot; (《三國志·卷一·魏書一·武帝紀》)
## 194 CE: Famine During the Puyang Campaign, ''Sanguozhi(2)''
##: '''English''': Cao Cao led his forces back and gave battle to Lü Bu at Puyang; his army fared ill and the two sides held their positions for over a hundred days. That year brought drought, locusts and scarcity of grain; the people ate each other. Lü Bu withdrew eastward to encamp at Shanyang.
##: '''Original:''' 太祖引军还，与布战于濮阳，太祖军不利，相持百余日。是时岁旱、虫蝗、少谷，百姓相食，布东屯山阳。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Lü Bu, Vol. 7&quot; (《三國志·卷七·魏書七·呂布臧洪傳》)
## 194 CE: Famine During the Puyang Campaign, ''Sanguozhi(3)''
##: '''English''': Cao Cao and Lü Bu held their positions at Puyang; Sima Lang thereupon led his household back to Wen. That year brought Great Famine; people ate each other. Lang gathered and succoured his kinsmen, tutored his younger brothers, and did not abandon his studies in that age of decline.
##: '''Original:''' 时岁大饥，人相食，朗收恤宗族，教训诸弟，不为衰世解业。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Sima Lang, Vol. 15&quot; (《三國志·卷十五·魏書十五·劉司馬梁張溫賈傳》)
## 194 CE: Famine During the Puyang Campaign, ''Hou Han Shu''
##: '''English''': Cao Cao heard of this and led his forces to attack Lü Bu; they fought repeatedly and held their positions for over a hundred days. That year brought drought, locusts and scarcity of grain; the people ate each other. Lü Bu withdrew to encamp at Shanyang.
##: '''Original:''' 曹操闻而引军击布，累战，相持百余日。是时，旱、蝗，少谷，百姓相食，布移屯山阳。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yan, Yuan Shu and Lü Bu, Vol. 75&quot; (《後漢書·卷七十五·劉焉袁術呂布列傳第六十五》)
# 194 CE: Cheng Yu's Human Jerky, Pei Songzhi's Commentary
#: '''English''': In the beginning, Cao Cao's forces lacked provisions. 
#: Cheng Yu seized supplies from his home county to provide three days' rations, mixed in no small part with dried human flesh. By this reason, he lost the favour of the ''(heavenly)'' court, and therefore never attained the rank of the Excellencies.
#: '''Original:''' 初，太祖乏食；昱略其本县，供三日粮，颇杂以人脯。由是失朝望，故位不至公。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Weijin Shiyu'', &quot;Biography of Cheng Yu, Vol. 14&quot; (裴松之《三國志注·卷十四·魏書十四·程昱傳》引《魏晉世語》)
# 195 CE: Great Famine at Chengshi, ''Sanguozhi''
#: '''English''': Cao Cao's forces were stationed at Chengshi. Great Famine; people ate each other.
#: '''Original:''' 太祖军乘氏，大饥，人相食。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Xun Yu, Vol. 10&quot; (《三國志·卷十·魏書十·荀彧荀攸賈詡傳》)
# 195 CE: The Siege of Dongjun, ''Hou Han Shu''
#: '''English''': [...] At first they still dug for rats and boiled sinew and hide; afterwards there was nothing left to eat.
#: The chief clerk reported that there were three dou of rice in the inner kitchen and requested it be made into gruel. Zang Hong said: &quot;How could I alone enjoy this?&quot; He had it made into thin porridge and distributed among all the troops.
#: He also slew all his beloved concubine to feed his officers and men. The officers and men all wept; none could raise their eyes to look upon him. Seventy or eighty men and women died lying upon one another; not one deserted or betrayed him.
#: '''Original:''' （东郡）初尚掘鼠，煮筋角，后无所复食，主簿启内厨米三斗，请稍为饘粥，洪曰：&quot;何能独甘此邪？&quot;使为薄糜，遍班士众。又杀其爱妾，以食兵将。兵将咸流涕，无能仰视。男女七八十人相枕而死，莫有离叛。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biography of Zang Hong, Vol. 58&quot; (《後漢書·卷五十八·虞詡等列傳》)
# 195 CE: The Siege of Dongjun, ''Zizhi Tongjian''
#: '''English''': At first they still dug for rats and boiled sinew and hide; afterwards there was nothing left to eat. 
#: The chief clerk reported only three sheng of rice in the inner kitchen and requested it be made into gruel. Zang Hong sighed: &quot;How could I alone enjoy this!&quot; He had it made into thin porridge and distributed among all the troops; he also slew his beloved concubine to feed his officers and men. 
#: The officers and men all wept; none could raise their eyes to look upon him. Seven or eight thousand men and women died lying upon one another; not one deserted or betrayed him.
#: '''Original:''' 初尚掘鼠煮筋角，后无可复食者。主簿启内厨米三升，请稍以为饘粥，臧洪叹曰：&quot;何能独甘此邪！&quot;使作薄糜，遍班士众，又杀其爱妾以食将士。将士咸流涕，无能仰视者。男女七八千人，相枕而死，莫有离叛者。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Hou Han Shu''
#: '''English''': [...] One hu of grain fetched fifty thousand coins; beans and wheat twenty thousand. People ate each other; white bones lay heaped in piles, and the stench of rot filled the roads. [...] After Li Jue and Guo Si turned upon each other and the Son of Heaven departed eastward, Chang'an stood empty for over forty days. The strong scattered; the weak ate each other. Within two or three years, not a human trace remained in Guanzhong.
#: '''Original:''' 自（李）傕、（郭）汜相攻，天子东归后，是时，谷一斛五十万，豆、麦二十万，人相食啖，白骨委积，臭秽满路。……长安城空四十余日，强者四散，蠃者相食，二三年间，关中无复人迹。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biography of Dong Zhuo, Vol. 72&quot; (《後漢書·卷七十二·董卓列傳第六十二》)
## 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Sanguozhi''
##: '''English''': At that time the three adjuncts still held several hundred thousand households. Li Jue and his confederates unleashed their troops to plunder, attacking and pillaging cities and towns. The people suffered Great Famine; within two years they had eaten each other to the last.
##: '''Original:''' 时三辅民尚数十万户，傕等放兵劫略，攻剽城邑，人民饥困，二年间相啖食略尽。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of Dong, the Two Yuans and Liu, Vol. 6&quot; (《三國志·卷六·魏書六·董二袁劉傳》)
## 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Jin Shu''
##: '''English''': [...] One hu of grain fetched fifty thousand coins; beans and wheat twenty thousand. People ate each other; white bones lay heaped in great mounds, the rotting remains befouling the roads. [...] Chang'an stood entirely empty; all scattered to the four winds. Within two or three years, not a traveller remained in Guanzhong. [...] Since Dong Zhuo's rebellion, the people had been scattered and adrift; grain reached over fifty thousand coins per shi, and many ate each other.
##: '''Original:''' 是时谷一斛五十万，豆麦二十万，人相食啖，白骨盈积，残骸余肉，臭秽道路。……长安城中尽空，并皆四散，二三年间，关中无复行人。……汉自董卓之乱，百姓流离，谷石至五十余万，人多相食。
##: '''Source:''' ''[[:w:Jin Shu|Jin Shu]]'', &quot;Treatise on Food and Commerce, Vol. 26&quot; (《晉書·卷二十六·志第十六·食貨》)
##195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Zizhi Tongjian''
##: '''English''': When Dong Zhuo first died, the three adjuncts still held several hundred thousand households. Li Jue and his confederates unleashed their troops to plunder; compounded by Great Famine, within two years the people had eaten each other nearly to the last. 
##: [...] At that time Chang'an stood empty for over forty days; the strong scattered, the weak ate each other, and within two or three years not a human trace remained in Guanzhong.
##: '''Original:''' 董卓初死，三辅民尚数十万户，李傕等放兵劫略，加以饥馑，二年间，民相食略尽。……是时，长安城空四十馀日，强者四散，羸者相食，二三年间，关中无复人迹。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# 195–197 CE: Wang Zhong the Cannibal, Pei Songzhi's Commentary
#: '''English''': Wang Zhong was a man of Fufeng who in his youth served as a village headman. When the three adjuncts fell into turmoil, Zhong, starving and desperate, ate human flesh, and followed a band of men southward toward Wuguan. [...]
#: The Master of the Wuguan Office, knowing that Zhong had once eaten human flesh, took him along on an imperial outing and had entertainers fasten a skull from a grave to Zhong's saddle, to the great amusement of all.
#: '''Original:''' 王忠，扶风人。少为亭长。三辅乱，忠饥乏噉人，随辈南向武关。……五官将知忠尝噉人，因从驾出行，令俳取冢间骷髅系著忠马鞍，以为欢笑。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Weilüe'', &quot;Annals of Emperor Wu, Vol. 1&quot; (裴松之《三國志注·魏書·武帝紀》引《魏略》)
# 196 CE: Liu Bei's Army Starves at Haixi, Zizhi Tongjian
#: '''English''': Liu Bei gathered his remaining forces and moved east to Guangling, gave battle to Yuan Shu, and was again defeated; he encamped at Haixi. Beset by hunger and hardship, his officers and men ate each other.
#: '''Original:''' 备收馀兵东取广陵，与袁术战，又败，屯于海西。饥饿困踧，吏士相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 62 (《資治通鑑·卷六十二》)
# 196 CE: Liu Bei's Army Starves at Haixi, Pei Songzhi's Commentary
#: '''English''': Liu Bei's army being at Guangling, hunger and hardship upon them; officers and men, high and low, ate each other.
#: '''Original:''' 備軍在廣陵，飢餓困踧，吏士大小自相啖食。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Yingxiong Ji'', &quot;Biography of the Progenitor Ruler, Vol. 32&quot; (裴松之《三國志注·卷三十二·蜀書·先主傳》引《英雄記》)
# 196 CE: Famine Under Gongsun Zan's Rule, ''Hou Han Shu''
#: '''English''': [...] That year brought drought and locusts; grain grew dear and people ate each other. Gongsun Zan, relying on his own abilities, showed no concern for the people.
#: '''Original:''' 是时，旱、蝗，谷贵，民相食。瓒恃其才力，不恤百姓。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yu, Gongsun Zan and Tao Qian, Vol. 73&quot; (《後漢書·卷七十三·劉虞公孫瓚陶謙列傳第六十三》)
# 197 CE: Famine Along the Yangtze and Huai, ''Hou Han Shu(1)''
#: '''English''': That year brought famine; along the Yangtze and Huai rivers, people ate each other.
#: '''Original:''' 是岁饥，江淮间民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Xian, Vol. 9&quot; (《後漢書·卷九·孝獻帝紀第九》)
## 197 CE: Famine Along the Yangtze and Huai, ''Hou Han Shu(2)''
##: '''English''': Yuan Shu's forces were weakened, his great generals dead, and his followers estranged and in revolt. Compounded by drought and failed harvests, his officers and people froze and starved; along the Yangtze and Huai, people had eaten each other nearly to the last.
##: '''Original:''' 术兵弱，大将死，众情离叛，加天旱岁荒，士民冻馁，江、淮间相食殆尽。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yan, Yuan Shu and Lü Bu, Vol. 75&quot; (《後漢書·卷七十五·劉焉袁術呂布列傳第六十五》)
## 197 CE: Famine Along the Yangtze and Huai, ''Sanguozhi''
##: '''English''': Yuan Shu's extravagance grew ever more excessive; his rear palace of several hundred consorts all wore fine silks, with surplus of grain and meat, whilst his officers and men froze and starved. Along the Yangtze and Huai the land was emptied; people ate each other.
##: '''Original:''' 荒侈滋甚，后宫数百皆服绮縠，余粱肉，而士卒冻馁，江淮间空尽，人民相食。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of Dong, the Two Yuans and Liu, Vol. 6&quot; (《三國志·卷六·魏書六·董二袁劉傳》)
## 197 CE: Famine Along the Yangtze and Huai, ''Zizhi Tongjian''
##: '''English''': Since the Zhongping era, the realm had fallen into turmoil; the people abandoned farming, armies rose on all sides, and provisions were ever wanting. When hungry, the troops plundered; when fed, they abandoned their surplus. Those who collapsed and scattered, undone by no enemy but themselves, were beyond counting. Yuan Shao in Hebei had his men subsist on mulberries; Yuan Shu along the Yangtze and Huai drew sustenance from cattail and river snails. The people ate each other, and the commanderies were left desolate.
##: '''Original:''' 中平以来，天下乱离，民弃农业，诸军并起，率乏粮谷，无终岁之计，饥则寇略，饱则弃馀，瓦解流离，无敌自破者，不可胜数。袁绍在河北，军人仰食桑椹。袁术在江淮，取给蒲蠃，民多相食，州里萧条。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 62 (《資治通鑑·卷六十二》)
# 238 CE: Siege of Xiangping. ''Sanguozhi''
#: '''English''': Gongsun Yuan was in dire stuation. His provisions exhausted, people ate each other, and the dead were very many.
#: '''Original:''' 渊窘急。粮尽，人相食，死者甚多。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of the Two Gongsuns, Tao and Four Zhangs, Vol. 8&quot; (《三國志·卷八·魏書八·二公孫陶四張傳》)
## 238 CE: Siege of Xiangping, ''Zizhi Tongjian''
##: '''English''': Gongsun Yuan was in dire situation; provisions in Xiangping were exhausted, people ate each other, and the dead were very many.
##: '''Original:''' 公孙渊窘急，粮尽，人相食，死者甚多。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 74 (《資治通鑑·卷七十四》)

==West Jin==

# 304 CE: The Famine of Chang'an and the Sack of Luoyang, ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': Shen Ju raised arms against Chang'an, yet was routed by (Sima) Yong. Zhang Fang greatly plundered Luo, then withdrew unto Chang'an. Thereupon the armies fell into dire want, and men did eat one another.
#: '''Original:''' 沈举举兵攻长安，为（司马）颙所败。张方大掠洛中，还长安。于是军中大馁，人相食。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals of Emperor Hui&quot; (《晋书·卷四·帝纪第四·惠帝》)
# 304 CE: The Plunder of Luoyang, in ''[[w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': (Zhang Fang) did seize from Luo above ten thousand bondsmen and bondswomen, both of state and private households, and marched them westward. The army, lacking victuals, did slay men and mingle their flesh with that of oxen and horses for sustenance.
#: '''Original:''' （张方）掠洛中官私奴婢万馀人而西。军中乏食，杀人杂牛马肉食之。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 85 (《资治通鉴》卷85)
# 306 CE: The Tyranny of Pan Tao and Bi Miao, in ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': (Pan) Tao and (Bi) Miao and their like seized (Sima) Yue and force him beyond the passes, falsely establishing a mobile administration, compelling the removal of ministers, issuing decrees by their own will, loosing soldiers to plunder and ravage, consuming the flesh of the common people, with corpses choking the roads and bleached bones filling the wilderness. Thus did the provincial lords betrayed their obligation, the cities and towns fall desolate, and the folk of Huai and Yu were casted into utter misery.
#: '''Original:''' （潘）滔、（毕）邈等劫（司马）越出关，矫立行台，逼徙公卿，擅为诏令，纵兵寇抄，茹食居人，交尸塞路，暴骨盈野。遂令方镇失职，城邑萧条，淮豫之萌，陷离涂炭。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biography of Zhou Jun et al.&quot; (《晋书·卷六十一·列传第三十一·周浚等》)
# 311 CE, eign of [[:w:Emperor Huai of Jin|Emperor Huai of Jin]]: The Rout at Ningping and the Death of Sima Yue, in ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': In the fifth year of Yongjia (the third month), (Sima) Yue did perish at Xiang. In the fourth month, Shi Le gave pursuit unto Ningping in Ku County; General Qian Duan sallied forth to resist him and fell in battle, the army breaking asunder. Thereupon Shi Le encircled the host of several hundred thousand with cavalry and loosed arrows upon them; the slain were heaped as mountains. Of princes, nobles, officers, and commoners, above a hundred thousand perished. Wang Mi's brother Zhang did burn the remnant and devour them. 
#: The people laid blame upon (Sima) Yue, and Emperor Huai issued a decree degrading Yue to the rank of a county king.
#: '''Original:''' 永嘉五年（三月），（司马越）薨于项。……（四月，）石勒追及于苦县宁平城，将军钱端出兵距勒，战死，军溃。……于是数十万众，（石）勒以骑围而射之，相践如山。王公士庶死者十余万。王弥弟璋焚其余众，并食之。天下归罪于（司马）越。（晋怀）帝发诏贬越为县王。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin (晉書)]]'', &quot;Biography of King Liang of Runan et al.&quot; (《晋书·卷五十九·列传第二十九·汝南王亮等》)
# 311 CE, Reign of [[:w:Emperor Huai of Jin|Emperor Huai of Jin]]: The Famine in the Passes, in ''[[w:Book of Jin|Book of Jin]](1)''
#: '''English''': At that time, famine ravaged the lands within the passes; the common folk consumed ate each other. Pestilence spreaded upon them, and bandits roamed openly, beyond the power of (Sima) Mo to suppress.
#: '''Original:''' 時關中饑荒，百姓相啖；加以疾疫，盜賊公行，（司马）模力不能制。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin (晉書)]]'', &quot;Biographies of the Imperial Clan&quot; (《晋书·卷三十七·列传第七·宗室》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](2)''
##: '''English''': Grand General Xun Xi memorialized to relocate the capital to Cangyuan; the Emperor was minded to comply, yet the great ministers, fearing (Pan) Tao, dared not carry out the edict, and the palace eunuchs, coveting their riches, were loath to depart. Famine grew great; people ate each other, and eight or nine in ten officials fled.
##: '''Original:''' 大将军苟晞表迁都仓垣,帝将从之,诸大臣畏滔,不敢奉诏,且宫中及黄门恋资财,不欲出。至是饥甚,人相食,百官流亡者十八九。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals, Emperor Huai &amp; Emperor Min&quot; (《晋书·卷五·帝纪第五·孝怀帝 孝愍帝》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](3)''
##: '''English''': By the Yongjia era, calamity and disorder had worsened greatly. East of Yongzhou, multitudes suffered hunger; they sold one another into bondage, and the wandering multitudes were beyond count. Six provinces — You, Bing, Si, Ji, Qin, and Yong — were struck by great locusts, devouring all grass, trees, and the fur of cattle and horses. Great pestilence followed, joined by famine. People were slain by brigands; corpses filled the rivers, and white bones covered the fields. As Liu Yao's forces pressed close, the court deliberated removing the capital to Cangyuan. People ate each other; famine and plague came together, and eight or nine in ten officials had fled.
##: '''Original:''' 至于永嘉,丧乱弥甚。雍州以东,人多饥乏,更相鬻卖,奔迸流移,不可胜数。幽、并、司、冀、秦、雍六州大蝗,草木及牛马毛皆尽。又大疾疫,兼以饥馑。百姓又为寇贼所杀,流尸满河,白骨蔽野。刘曜之逼,朝廷议欲迁都仓垣。人多相食,饥疫总至,百官流亡者十八九。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Treatise on Food and Commerce&quot; (《晋书·卷二十六·志第十六·食货》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](3)''
##: '''English''': Emperor Huai being besieged by Liu Yao, the imperial armies suffered repeated defeat, the treasury was exhausted, and the hundred officials were greatly famished; smoke of cooking fires was seen in no house. The starving fed upon one another. In the west, where Emperor Min resided, hunger was exceeding great; a peck of grain cost two taels of gold, and more than half the people perished.
##: '''Original:''' 怀帝为刘曜所围,王师累败,府帑既竭,百官饥甚,比屋不见火烟,饥人自相啖食。愍皇西宅,馁馑弘多,斗米二金,死者太半。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Treatise on Food and Commerce&quot; (《晋书·卷二十六·志第十六·食货》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](4)''
##: '''English''': When Luoyang fell into chaos, with thieves running rampant, people ate each other out of hunger. (Zhi) Yu, being ever poor and frugal, perished at last of starvation.
##: '''Original:''' 及洛京荒乱,盗窃纵横,人饥相食。虞素清贫,遂以馁卒。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Huangfu Mi et al.&quot; (《晋书·卷五十一·列传第二十一·皇甫谧等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](5)''
##: '''English''': (Wang) Mi, together with (Liu) Yao, attacked Xiangcheng and pressed upon the capital. The capital suffered a Great Famine; people ate each other, the common folk fled, and the dukes and ministers escaped to Heyin.
##: '''Original:''' 王弥后与曜寇襄城,遂逼京师。时京邑大饥,人相食,百姓流亡,公卿奔河阴。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Wang Mi et al.&quot; (《晋书·卷一百·列传第七十·王弥等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](6)''
##: '''English''': Wang Mi and Liu Yao arrived and joined (Huyan) Yan in besieging Luoyang. Within the city, famine was dire; people ate each other, the hundred officials scattered, and none held firm. The Xuanyang Gate fell; Mi and Yan entered the Southern Palace, ascended the Taiji Front Hall, and loosed their soldiers in great plunder, seizing all palace women and treasures. Yao thereupon slew all the princes, nobles, and officers below, in which numbered more than thirty thousand in all, and thereupon raised a great mound of their skulls north of the Luo River.
##: '''Original:''' 王弥、刘曜至,复与晏会围洛阳。时城内饥甚,人皆相食,百官分散,莫有固志。宣阳门陷,弥、晏入于南宫,升太极前殿,纵兵大掠,悉收宫人、珍宝。曜于是害诸王公及百官已下三万余人,于洛水北筑为京观。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Ere long, Luoyang fell to famine and distress; people ate each other, and eight or nine in ten officials had fled.
##: '''Original:''' 既而洛阳饥困,人相食,百官流亡者什八九。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 87 (《资治通鉴》卷87)
# 311 CE, Reign of Emperor Huai of Jin (永嘉五年): Great Famine and Cannibalism After the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': When Luoyang fell, Grand Commandant Xun Fan fled to Yangcheng, and General of the Guard Hua Hui fled to Chenggao. A Great Famine prevailed; the bandit chief Hou Du and his ilk seized men for food, and many of Fan's and Hui's followers were thus devoured.
#: '''Original:''' 及洛阳不守，太尉荀藩奔阳城，卫将军华荟奔成皋。时大饥，贼帅侯都等每略人而食之，藩、荟部曲多为所啖。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Shao Xu et al.&quot; (《晋书·卷六十三·列传第三十三·邵续等》)
# 312 CE: Cannibalism Among Han Zhao Troops, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': The Han Zhao generals Zhao Gu and Wang Sang, fearing absorption by Shi Le, sought to lead their forces back to Pingyang. Provisions within the army ran short, and soldiers ate each other.
#: '''Original:''' 汉安北将军赵固、平北将军王桑恐为石勒所并，欲引兵归平阳。军中乏粮，士卒相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 88 (《资治通鉴》卷88)
# 312 CE: Cannibalism in Shi Le's Army at Gepei, in ''[[:w:Book of Jin|Book of Jin]]'' and ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': (Shi) Le, at Gepei, built dwellings, encouraged farming, and constructed boats, intending to attack Jiankang. Yet wherever he marched, the people had fortified their walls and cleared the fields; nothing could be plundered, and great famine fell upon the army, so that soldiers ate each other.
#: '''Original:''' 勒于葛陂缮室宇，课农造舟，将寇建邺。……勒所过路次，皆坚壁清野，采掠无所获，军中大饥，士众相食。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Shi Le I&quot; (《晋书·卷一百四·载记第四·石勒上》)
# 312 CE: Cannibalism in Shi Le's Army at Gepei, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': As Shi Le marched north from Gepei, all along his path the people had fortified and cleared the fields; nothing could be seized. Famine within the army grew dire, and soldiers ate each other.
#: '''Original:''' 石勒自葛陂北行，所过皆坚壁清野，虏掠无所获，军中饥甚，士卒相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 88 (《资治通鉴》卷88)
# 314 CE: Monstrous Birth and Cannibalism in Guangyi, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': The wife of Yang Chong of Guangyi bore a child with two heads; her brother stole and ate it, and died within three days.
#: '''Original:''' 光义人羊充妻产子二头，其兄窃而食之，三日而死。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)
# 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Book of Jin|Book of Jin]](1)''
#: '''English''': In the tenth month of winter, the capital Chang'an suffered dire famine; a peck of grain cost two taels of gold, people ate each other, and more than half perished.
#: '''Original:''' 冬十月，京师饥甚，米斗金二两，人相食，死者太半。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals, Emperor Huai &amp; Emperor Min&quot; (《晋书·卷五·帝纪第五·孝怀帝 孝愍帝》)
## 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Book of Jin|Book of Jin]](2)''
##: '''English''': When Liu Yao again besieged the capital, (Suo) Chen and Qu Yun held fast to the inner city of Chang'an. Within, famine was dire; people ate each other, and the dead, fugitives, and deserters were beyond restraint; only the thousand loyal troops from Liangzhou stood firm unto death.
##: '''Original:''' 后刘曜又率众围京城、綝与麹允固守长安小城。……城中饥窘，人相食，死亡逃奔不可制，唯凉州义众千人守死不移。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Xie Xi et al.&quot; (《晋书·卷六十·列传第三十·解系等》)
## 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': In the eighth month, the Han Zhao Grand Marshal (Liu) Yao pressed upon Chang'an. Yao stormed the outer city; Qu Yun and Suo Chen withdrew to defend the inner city. All communication within and without was severed; famine within grew dire. A peck of grain cost two taels of gold, people ate each other, and more than half had perished; deserters and fugitives could not be restrained. Only the thousand loyal troops from Liangzhou stood firm. In the imperial granary there remained but several dozen cakes of leaven; Qu Yun ground them into gruel to feed the Emperor, yet ere long even these were exhausted.
##: '''Original:''' 八月，汉大司马曜逼长安。……曜攻陷長安外城，麴允、索綝退保小城以自固。內外斷絕，城中饑甚。斗米值金二兩，人相食，死者大半，亡逃不可制。唯涼州義眾千人守死不移。太倉有麴數十餅，麴允屑之為粥以供帝，既而亦盡。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 89 (《资治通鉴》卷89)
# 316 CE: Great Famine and Cannibalism in Beidi, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': Famine in Beidi was dire; people ate each other. Qiang Qiou's army transported grain to supply Qu Chang, but was defeated by Liu Ya.
#: '''Original:''' 北地饥甚，人相食啖，羌酋大军须运粮以给麹昌，刘雅击败之。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', Vol. 102 &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)

==East Jin==
# 319 CE: Slicing and Eating of Du Zeng's Flesh, ''Book of Jin''
#: '''English''': Du Zeng's forces collapsed; his generals Ma Jun and Su Wen captured him and surrendered to Zhou Fang. Zhou Fang wished to bring him alive to Wuchang, but Zhu Gui's son Zhu Chang and Zhao You's son Zhao Yin both begged for Du Zeng to avenge their fathers' grievances. Du Zeng was thereupon beheaded; Chang and Yin sliced his flesh and ate it.
#: '''Original:''' 曾众溃，其将马俊、苏温等执曾诣访降。访欲生致武昌，而朱轨息昌、赵诱息胤皆乞曾以复冤，于是斩杜曾，而昌、胤脔其肉而啖之。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 100, &quot;Biographies, Vol. 70: Wang Mi et al.&quot; (《晋书·卷一百·列传第七十·王弥等》)
# c. 321 CE: Xu Kan Fed to His Own Kin After Execution, ''Book of Jin''
#: '''English''': Shi Jilong attacked and captured Xu Kan, sending him to Xiangguo. Shi Le had him bagged and hurled to his death from the hundred-foot tower, then ordered the wives and children of Bu Du and others to disembowel and eat him; three thousand of Xu Kan's surrendered troops were buried alive.
#: '''Original:''' 石季龙攻陷徐龛，送之襄国，勒囊盛于百尺楼自上扑杀之，令步都等妻子刳而食之，坑龛降卒三千。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 105, &quot;Chronicles, Vol. 5: Shi Le, Part II et al.&quot; (《晋书·卷一百五·载记第五·石勒下等》)
# c. 337 CE: Shi Sui Slays Palace Women and Nuns, ''Book of Jin(1)''
#: '''English''': After Shi Sui assumed full governance, he abandoned himself to wine and lust, acting with arrogant depravity. He would roam the fields with music playing as he entered, or venture by night into the homes of court officials to violate their wives and concubines. 
#: Of the palace women whom he had adorned and found comely, he would behead them, wash away the blood, place their heads upon platters, and pass them round for viewing. He also brought in comely Buddhist nuns, defiled them, then slew them; their flesh was boiled together with beef and mutton and eaten, and portions were also distributed to his attendants, who were interested in the flavor.
#: '''Original:''' 邃自总百揆之后，荒酒淫色，骄恣无道，或盘游于田，悬管而入，或夜出于宫臣家，淫其妻妾。妆饰宫人美淑者，斩首洗血，置于盘上，传共视之。又内诸比丘尼有姿色者，与其交亵而杀之，合牛羊肉煮而食之，亦赐左右，欲以识其味也。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 106, &quot;Chronicles, Vol. 6: Shi Jilong, Part I&quot; (《晋书·卷一百六·载记第六·石季龙上》)
## c. 337 CE: Shi Sui Slays and Cooks Palace Women and Nuns, ''Zizhi Tongjian''
##: '''English''': Shi Sui, Crown Prince of Later Zhao, was arrogant, lustful, and cruel; he delighted in adorning comely consorts, beheading them, washing away the blood, placing their heads upon platters, and passing them amongst his guests for viewing. He further cooked their flesh and shared it for eating.
##: '''Original:''' 邃骄淫残忍，好妆饰美姬，斩其首，洗血置盘上，与宾客传观之，又烹其肉共食之。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 95 (《资治通鉴》卷95)
# 351 CE: Great Famine in Si and Ji Provinces, ''Book of Jin(1)''
#: '''English''': Bandits and rebels arose like swarms; a Great Famine struck Si and Ji Provinces; people ate each other. 
#: From the final years of Shi Jilong, Ran Min had dispersed all the granaries and treasuries to cultivate personal loyalty. Warfare with the Qiang and Hu raged without cease, with battles every month. 
#: The transplanted households of Qing, Yong, You, and Jing Provinces, together with the Di, Qiang, Hu, and Man peoples, numbering several hundred myriads, returned to their native lands; their routes met in one point, where all of they slaughtered and plundered one another. With famine and pestilence, only two or three in ten reached their destinations. Throughout the realm there was great disorder, and none remained to till the fields.
#: '''Original:''' 贼盗蜂起，司、冀大饥，人相食。自季龙末年而闵尽散仓库以树私恩。与羌胡相攻，无月不战。青、雍、幽、荆州徙户及诸氐、羌、胡、蛮数百余万，各还本土，道路交错，互相杀掠，且饥疫死亡，其能达者十有二三。诸夏纷乱，无复农者。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 107, &quot;Chronicles, Vol. 7: Shi Jilong, Part II&quot; (《晋书·卷一百七·载记第七·石季龙下》)
## 351 CE: Great Famine in Si and Ji Provinces, ''Zizhi Tongjian''
##: '''English''': The several hundred myriad transplanted peoples of Qing, Yong, You, and Jing Provinces — along with the Di, Qiang, Hu, and Man — whom Later Zhao had relocated, found the laws of Zhao no longer enforced and each returned to their native lands.
##: Their routes met in one point, where all of they slaughtered and plundered one another; only two or three in ten reached their destinations. The Central Plains fell into great disorder. Famine and pestilence followed; people ate each other, and none remained to till the fields.
##: '''Original:''' 后赵所徙青、雍、幽、荆四州人民及氐、羌、胡蛮数百万口，以赵法禁不行，各还本土；道路交错，互相杀掠，其能达者什有二、三。中原大乱。因以饥疫，人相食，无复耕者。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 99 (《资治通鉴》卷99)
# 352 CE: Famine in Ye, ''Book of Jin''
#: '''English''': Famine struck Ye; people ate each other. The palace women from the time of Shi Jilong were nearly all consumed.
#: '''Original:''' 邺中饥，人相食，季龙时宫人被食略尽。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 107, &quot;Chronicles, Vol. 7: Shi Jilong, Part II&quot; (《晋书·卷一百七·载记第七·石季龙下》)
## 352 CE: Famine in Ye'', Zizhi Tongjian''
##: '''English''': A Great Famine struck Ye; people ate each other. The palace women from the time of the former Zhao were nearly all consumed.
##: '''Original:''' 邺中大饥，人相食，故赵时宫人被食略尽。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 99 (《资治通鉴》卷99)
# 356 CE: Siege of Duan Kan's City, ''Zizhi Tongjian''
#: '''English''': Duan Kan defended the Yin city under siege; the roads for gathering firewood were cut off, and people ate each other within the city.
#: '''Original:''' 段龛婴城自守，樵采路绝，城中人相食。
#: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 100 (《资治通鉴·卷一百》)
# 385 CE: Great Famine at Chang'an, ''Book of Jin''
#: '''English''': At this time there was a Great Famine in Chang'an; people ate each other, and the generals, upon returning home, spat out flesh to feed their wives and children.
#: '''Original:''' 时长安大饥，人相食，诸将归而吐肉以饴妻子。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
## 385 CE: Great Famine at Chang'an, ''Wei Shu''
##: '''English''': Great Famine in Chang'an; people ate each other. Yao Chang rebelled at Beidi and allied with [Murong] Chong, jointly attacking Chang'an.
##: '''Original:''' 长安大饥，人民相食。姚苌叛于北地，与冲连和，合攻长安。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 95 &quot;Biographies 83, Liu Cong of the Xiongnu et al.&quot; (《魏书·卷九十五·列传第八十三·匈奴刘聪等》)
## 385 CE: Great Famine at Chang'an, ''Zizhi Tongjian''
##: '''English''': In the first month, [Former] Qin's [Fu] Jian held a banquet for his ministers. Chang'an was then stricken by famine; people ate each other, and the generals, upon returning home, spat out flesh to feed their wives and children.
##: '''Original:''' 正月，(前)秦(苻)堅朝饗群臣，時長安飢，人相食，諸將歸，吐肉以飼妻子。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 106 (《资治通鉴·卷一百零六》)
# 385 CE: Murong Chong's Forces Eat the Slain, ''Book of Jin''
#: '''English''': [Murong] Chong further dispatched his Secretariat Director Gao Gai to lead troops in a night assault on Chang'an, breaching the southern gate and entering the southern city. General of the Left Dou Chong and General of the Front Guards Li Bian and others repelled them, beheading 1,800 men, and divided the corpses for consumption.
#: '''Original:''' （慕容）冲又遣其尚书令高盖率众夜袭长安，攻陷南门，入于南城。左将军窦冲、前禁将军李辩等击败之，斩首千八百级，分其尸而食之。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
# 385 CE: Famine in You and Ji Prefectures, ''Book of Jin''
#: '''English''': Murong Gui's troops suffered greatly from hunger and many fled to Zhongshan; the people of You and Ji prefectures ate each other. Earlier, a popular rhyme in the Pass East had said: &quot;Youzhou — born to be destroyed; if not destroyed, the people shall be extinguished.&quot; This was [Murong] Cui's birth name. Having held out against [Fu] Pi for a full year, the common people were nearly all dead.
#: '''Original:''' 慕容垂军人饥甚，多奔中山，幽、冀人相食。初，关东谣曰：&quot;幽州，生当灭。若不灭，百姓绝。&quot;（慕容）垂之本名。与（符）丕相持经年，百姓死几绝。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
## 385 CE: Famine in You and Ji Prefectures, ''Zizhi Tongjian''
##: '''English''': Yan and Qin having held out against each other for a full year, You and Ji prefectures suffered a Great Famine; people ate each other, and settlements lay desolate. Many of Yan's soldiers starved to death; the King of Yan, [Murong] Cui, forbade the people from raising silkworms and had them subsist on mulberry berries.
##: '''Original:''' 燕、秦相持經年，幽、冀大饑，人相食，邑落蕭條，燕之軍士多餓死，燕王（慕容）垂禁民養蠶，以桑椹為食。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 106 (《资治通鉴·卷一百零六》)
# 386 CE: Fu Deng's Army Eats the Slain, ''Book of Jin''
#: '''English''': [Fu] Deng, having succeeded Wei Ping, thenceforth held sole command of military campaigns. At this time drought brought widespread hunger, and the roads were lined with the starving dead. Whenever Deng won a battle and slew the enemy, he called it &quot;cooked meat,&quot; and said to his men: &quot;You fight in the morning and by evening are sated with flesh — why fear hunger!&quot; The troops followed his lead, eating the flesh of the slain, and were thereby well-fed and fit for battle.
#: '''Original:''' （苻）登既代卫平，遂专统征伐。是时岁旱众饥，道殣相望，登每战杀贼，名为熟食，谓军人曰：&quot;汝等朝战，暮便饱肉，何忧于饥！&quot;士众从之，啖死人肉，辄饱健能斗。 
#: '''Source:''' [[wikipedia:Book of Jin|''Book of Jin'']], Vol. 115 &quot;Chronicles 15, Fu Pi et al.&quot; (《晋书·卷一百十五·载记第十五·苻丕等》)
# 387 CE: Famine in Jiuquan, ''Book of Jin''
#: '''English''': Wang Mu seized Jiuquan by surprise and proclaimed himself General-in-Chief and Governor of Liangzhou. At this time grain prices soared; one dou fetched five hundred cash, people ate each other, and more than half perished.
#: '''Original:''' 王穆袭据酒泉，自称大将军、凉州牧。时谷价踊贵，斗直五百，人相食，死者太半。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 122 &quot;Chronicles 22, Lü Guang et al.&quot; (《晋书·卷一百二十二·载记第二十二·吕光等》)
# 387 CE: Famine in Liangzhou, ''Zizhi Tongjian''
#: '''English''': Great Famine in Liangzhou; one dou of rice fetched five hundred cash, people ate each other, and more than half perished.
#: '''Original:''' 涼州大饑，米斗直錢五百，人相食，死者太半。
#: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 112 (《资治通鉴·卷第一百一十二》)
# c. 399 CE: Sun En Rebellion, ''Song Shu''
#: '''English''': In this time all means of livelihood were exhausted and the weak and elderly were many; the eastern lands suffered famine, and people exchanged children to eat. 
#: '''Original:''' 时生业已尽，老弱甚多，东土饥荒，易子而食;
#: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 100 &quot;Biographies 60, Preface&quot; (《宋书·卷一百·列传第六十·自序》)
## c. 399 CE: Sun En Rebellion, ''Wei Shu''
##: '''English''': When [Sun] En raised his rebellion, all eight commanderies became a field of carnage. … The rebels' prohibitions went unheeded; they killed at will, and the number of officers and commoners slain was beyond reckoning. Some county magistrates were pickled and fed to their own wives and children; those who refused were dismembered. Such was their cruelty.
##: '''Original:''' （孙）恩既作乱，八郡尽为贼场，……贼等禁令不行，肆意杀戮，士庶死者不可胜计，或醢诸县令以食其妻子，不肯者辄支解之，其虐如此。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 96 &quot;Biographies 84, the Usurper Jin's Sima Rui et al.&quot; (《魏书·卷九十六·列传第八十四·僭晋司马叡等》)
# 401 CE, Longan 5: Omen of Famine and Usurpation, ''Book of Jin''
#: '''English''': Huan Xuan's memorial arrived, defying imperial intent and affronting the throne. Thereafter Xuan usurped the throne, threw the capital into disorder; there was a Great Famine, people ate each other, and the common people fled — all were fulfillments of these omens.
#: '''Original:''' 九月，桓玄表至，逆旨陵上。其后玄遂篡位，乱京都，大饥，人相食，百姓流亡，皆其应也。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 13 &quot;Treatises 3, Astronomy III&quot; (《晋书·卷十三·志第三·天文下》)
# 402 CE: Famine at Guzang, ''Book of Jin''
#: '''English''': Grain prices at Guzang soared; one dou fetched five thousand cash, people ate each other, and over a hundred thousand starved to death. The city gates were shut by day, and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the barbarians. [Lü] Long, fearing this would demoralize the populace, had them all buried alive, corpses piled up and filled the streets.
#: '''Original:''' 姑臧谷价踊贵，斗直钱五千文，人相食，饿死者十余万口。城门昼闭，樵采路绝，百姓请出城乞为夷虏奴婢者日有数百。隆惧沮动人情，尽坑之，于是积尸盈于衢路。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 122 &quot;Chronicles 22, Lü Guang et al.&quot; (《晋书·卷一百二十二·载记第二十二·吕光等》)
## 402 CE: Famine at Guzang, ''Wei Shu''
##: '''English''': Juqu Mengxun and Tufa Rutan attacked repeatedly, leaving the people of Hexi unable to farm to the west. Grain prices soared; one dou fetched five thousand cash, people ate each other, and over a thousand starved to death. The city gates of Guzang were shut by day and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the barbarians. [Lü] Long, fearing this would demoralize the populace, had them all buried alive.
##: '''Original:''' 沮渠蒙逊、秃发辱檀频来攻击，河西之民，不得农西，谷价涌贵，斗直钱五千文，人相食，饿死者千余口。姑臧城门昼闭，樵采路断，民请出城，乞为夷虏奴婢者，日有数百。隆恐沮动人情，尽坑之。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 95 &quot;Biographies 83, Liu Cong of the Xiongnu et al.&quot; (《魏书·卷九十五·列传第八十三·匈奴刘聪等》)
## 402 CE: Famine at Guzang, ''Zizhi Tongjian''
##: '''English''': Great Famine at Guzang; one dou of rice fetched five thousand cash, people ate each other, and over a hundred thousand starved to death. The city gates were shut by day, and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the Hu barbarians; Lü Long, loathing the effect on morale, had them all buried alive, corpses piled up and filled the roads.
##: '''Original:''' 姑臧大饥，米斗直钱五千，人相食，饥死者十馀万口。城门昼闭，樵采路绝，民请出城为胡虏奴婢者，日有数百，吕隆恶其沮动众心，尽坑之，积尸盈路。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 112 (《资治通鉴·卷一百一十二》)
# 402 CE: Astronomical Omen of Famine, ''Book of Jin''
#: '''English''': In the fourth month, on the day xinsi, the moon occluded Mercury. In the seventh month, Great Famine; people ate each other.
#: '''Original:''' 元兴元年四月辛丑，月奄辰星。七月，大饥，人相食。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 12 &quot;Treatises 2, Astronomy II&quot; (《晋书·卷十二·志第二·天文中》)
## 402 CE: Famine in the Eastern Regions, ''Book of Jin(1)''
##: '''English''': In the seventh month of Yuanxing 1, Great Famine; people ate each other. Six or seven in ten east of the Zhe River died or fled; the population of Wu Commandery and Wuxing was halved, and tens of thousands more fled westward.
##: '''Original:''' 元兴元年七月，大饥，人相食。浙江以东流亡十六七，吴郡、吴兴户口减半，又流奔而西者万计。
##: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 13 &quot;Treatises 3, Astronomy III&quot; (《晋书·卷十三·志第三·天文下》)
## 402 CE: Famine in the Eastern Regions, ''Song Shu''
##: '''English''': In the seventh month [of Yuanxing 1], Great Famine; people ate each other. Six or seven in ten east of the Zhe River starved to death or fled; the population of Wu Commandery and Wuxing was halved, and tens of thousands more fled westward.
##: '''Original:''' （元兴元年）七月，大饥，人相食。浙江东饿死流亡十六七，吴郡、吴兴户口减半；又流奔而西者万计。
##: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 25 &quot;Treatises 15, Astronomy III&quot; (《宋书·卷二十五·志第十五·天文三》)
# 402 CE Kong Clan Distributes Grain, ''Song Shu''
#: '''English''': After the Sun En rebellion, the eastern lands suffered famine; people ate each other. The Kong clan distributed their household grain to relieve the neighbourhood, saving many lives; those who bore children thereafter named them Kong in gratitude.
#: '''Original:''' 及孙恩乱后，东土饥荒，人相食，孔氏散家粮以赈邑里，得活者甚众，生子皆以孔为名焉。
#: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 81 &quot;Biographies 41, Liu Xiuzhi et al.&quot; (《宋书·卷八十一·列传第四十一·刘秀之等》)
## 402 CE: Kong Clan Distributes Grain, ''Nan Shi''
##: '''English''': After the Sun En rebellion, the eastern lands suffered famine; people ate each other. The Kong clan distributed their household grain to relieve the neighbourhood, saving many lives; those who bore children thereafter named them Kong in gratitude.
##: '''Original:''' 孙恩乱后，东土饥荒，人相食，孔氏散家粮以振邑里，得活者甚众，生子皆以孔为名焉。
##: '''Source:''' [[:w:Nan Shi|''Nan Shi'']], Vol. 35 &quot;Biographies 25, Liu Zhan et al.&quot; (《南史·卷三十五·列传第二十五·刘湛等》)
# 409 CE: Cannibalism as Punishment for Regicide, ''Bei Shi''
#: '''English''': [Tuoba] Shao, together with several attendants and eunuchs, scaled the palace walls and violated the forbidden precinct. The Emperor [Daowu of Northern Wei, Tuoba Gui] started up in alarm, reached for his bow and sword but could not find them, and died suddenly. … The guards seized and delivered Shao. Thereupon Shao and his mother were put to death; the attending eunuchs and palace women who had acted as inner accomplices, numbering over ten, were executed. Those who had first laid hands upon the imperial person were carved alive and eaten by the assembled ministers on the main avenue south of the city.
#: '''Original:''' （拓跋）绍乃与帐下及宦者数人逾宫犯禁。帝（北魏道武皇帝拓跋珪）惊起，求弓刀不及，暴崩。……卫士执送绍，于是赐绍母子死，诛帐下阉官、宫人为内应者十数人。其先犯乘舆者，群臣于城南都街生脔食之。
#: '''Source:''' [[:w:Bei Shi|''Bei Shi'']], Vol. 16 &quot;Biographies 4, The Seven Princes of Daowu et al.&quot; (《北史·卷十六·列传第四·道武七王等》)
## 409 CE: Cannibalism as Punishment for Regicide, ''Wei Shu''
##: '''English''': The Supreme Ancestor (Taizong) arrived at the west of the city; the guards seized and delivered Shao. Thereupon Shao and his mother were put to death; the attending eunuchs and palace women who had acted as inner accomplices, numbering over ten, were executed. Those who had first laid hands upon the imperial person were carved alive and eaten by the assembled ministers on the main avenue south of the city.
##: '''Original:''' 太宗至城西，卫士执送绍。于是赐绍母子死，诛帐下阉官、宫人为内应者十数人，其先犯乘舆者，群臣于城南都街生脔割而食之。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 16 &quot;Biographies 4, The Seven Princes of Daowu&quot; (《魏书·卷十六·列传第四·道武七王》)
## 409 CE: Cannibalism as Punishment for Regicide, ''Zizhi Tongjian''
##: '''English''': Those who had first laid hands upon the imperial person [Tuoba Gui] were carved and eaten by the assembled ministers.
##: '''Original:''' 其先犯乘舆（拓跋珪）者，群臣脔食之。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']] (《资治通鉴》)


==南北朝==

# 431 CE: Siege of Nan'an, ''Bei Shi''
#: '''English''': Helian Ding dispatched Wei Dai, Duke of Beiping, with ten thousand men to attack Nan'an. Within the city there was Great Famine; people ate each other.
#: '''Original:''' 赫连定遣其北平公韦代率众万人攻南安。城内大饥，人相食。
#: '''Source:''' [[:w:Bei Shi|Bei Shi]], Vol. 93 &quot;Biographies, 81: Pretenders and Vassals&quot; (《北史·卷九十三·列传第八十一·僭伪附庸》)
# 431 CE: Siege of Nan'an, ''Book of Wei''
#: '''English''': Helian Ding dispatched Wei Dai, Duke of Beiping, with ten thousand men to attack Nan'an. Within the city there was Great Famine; people ate each other.
#: '''Original:''' 赫连定遣其北平公韦代率众一万攻南安，城内大饥，人相食。
#: '''Source:''' [[:w:Wei Shu|Wei Shu]], Vol. 99 &quot;Biographies, 87: Zhang Shi, Governor of Liangzhou et al.&quot; (《魏书·卷九十九·列传第八十七·凉州牧张实等》)
# 431 CE: Siege of Nan'an, ''Zizhi Tongjian''
#: '''English''': The Xia ruler (Helian Ding) attacked and defeated the Qin general Yao Xian; thereupon he dispatched his uncle Wei Fa, Duke of Beiping, with ten thousand men to attack Nan'an. Within the city there was Great Famine; people ate each other.
#: '''Original:''' 夏主（赫连定）击秦将姚献，败之；遂遣其叔父北平公韦伐帅众一万攻南安。城中大饥，人相食。
#: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 122 (《资治通鉴》卷122)
# Yuanjia Era: Medicinal Corpse, ''Yi Yuan''
#: '''English''': In the Yuanjia era, the Hu family of Yuzhang opened the tomb of [[:w:Marquis of Haihun | King Changyi]], and a man of Qingzhou opened the tomb of [[:w:Duke Xiang of Qi|Duke Xiang of Qi]]; both found golden hooks, whilst the corpses remained intact in the rocks. This may not be certain, yet the corpse of [[:w:Jing Fang|Jing Fang]] remained complete until the Yixi era; the flesh of such frozen corpses was fit for medicine, and soldiers carved and ate thereof.
#: '''Original:''' 元嘉中，豫章胡家奴開邑王冢，青州人開齊襄公冢，並得金鉤，而屍骸露在岩中儼然。茲亦未必有憑而然也，京房屍至義熙中猶完具，殭屍人肉堪為藥，軍士分割食之。
#: '''Source:''' [[:w:zh:异苑|Yi Yuan]] by Liu Jingshu (《异苑》)
# 441 CE: Siege of Jiuquan, ''Book of Song''
#: '''English''': In the seventh month, Tuoba Tao dispatched an army to besiege Jiuquan. In the tenth month, there was famine within the city and ten thousand people starved to death; Juqu Tianzhou killed his wife to feed the soldiers. When the food was exhausted, the city fell; Tianzhou was captured and taken to Pingcheng, where he was executed.
#: '''Original:''' 七月，拓跋焘遣军围酒泉。十月，城中饥，万余口皆饿死，（沮渠）天周杀妻以食战士；食尽，城乃陷，执天周至平城，杀之。
#: '''Source:''' [[:w:Song Shu|Song Shu]], Vol. 98 &quot;Biographies, 58: Di Hu&quot; (《宋书·卷九十八·列传第五十八·氐胡》)
# 441: Siege of Jiuquan, ''Zizhi Tongjian''
#: '''English''': Food was exhausted within the city of Jiuquan and ten thousand people starved to death; Juqu Tianzhou killed his wife to feed the soldiers.
#: '''Original:''' 酒泉城中食尽，万馀口皆饿死，沮渠天周杀妻以食战士。
#: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 123 (《资治通鉴》卷123)
# c. 450: Qingzhou Famine, ''Book of Southern Qi''
#: '''English''': At the end of the Yuanjia era, there was famine in Qingzhou; people ate each other.
#: '''Original:''' 元嘉末，青州饥荒，人相食。
#: '''Source:''' [[:w:Nan Qi Shu|Book of Southern Qi]], Vol. 28 &quot;Biographies, 9: Cui Zushi et al.&quot; (《南齐书·卷二十八·列传第九·崔祖思等》)
# c. 450: Qingzhou Famine, ''Nan Shi''
#: '''English''': At the end of the Yuanjia era, there was famine in Qingzhou; people ate each other. (Liu) Shanming had stored grain; he himself ate only thin porridge and opened his granaries to provide relief, whereby many in the village were saved. The people thereafter called his fields the &quot;Life-Sustaining Fields.&quot;
#: '''Original:''' 元嘉末，青州饥荒，人相食。（刘）善明家有积粟，躬食饘粥，开仓以救，乡里多获全济，百姓呼其家田为续命田。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 49 &quot;Biographies, 39: Yu Gaozhi et al.&quot; (《南史·卷四十九·列传第三十九·庾杲之等》)
# 453: Execution of Zhang Chaozhi, ''Song Shu''
#: '''English''': Zhang Chaozhi, hearing the troops had entered, fled to the old foundations of the He-dian hall and stopped at the site of the imperial bed, where he was killed by rebel soldiers. They cut open his intestines, gouged out his heart, and carved his flesh; the generals ate it raw and burned his skull.
#: '''Original:''' 张超之闻兵入，遂走至合殿故基，正于御床之所，为乱兵所杀。割肠刳心，脔剖其肉，诸将生啖之，焚其头骨。
#: '''Source:''' [[:w:Song Shu|Song Shu]], Vol. 99 &quot;Biographies, 59: Two Villains&quot; (《宋书·卷九十九·列传第五十九·二凶》)
# 453: Execution of Zhang Chaozhi, ''Nan Shi''
#: '''English''': Zhang Chaozhi fled to the site of the imperial bed in the He-dian hall. He was killed by soldiers; they gouged his intestines and heart, carved his flesh, and the generals ate it raw. They burned his skull.
#: '''Original:''' 张超之走至合殿御床之所。为军士所杀，刳肠割心，诸将脔其肉，生啖之。焚其头骨。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 14 &quot;Biographies, 4: Song Imperial Clan and Princes&quot; (《南史·卷十四·列传第四·宋宗室及诸王下》)
# 453: Execution of Zhang Chaozhi, ''Zizhi Tongjian''
#: '''English''': Zhang Chaozhi fled to the site of the imperial bed in the He-dian hall. He was killed by soldiers; they gouged his intestines and heart, and the generals carved his flesh and ate it raw.
#: '''Original:''' 张超之走至合殿御床之所。为军士所杀，刳肠割心，诸将脔其肉，生啖之。
#: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 127 (《资治通鉴》卷127)
# c. 454 CE: Liu Yong's Consumption of Scabs, ''Book of Song''
#: '''English''': Liu Yong had a passion for eating scabs, believing the taste resembled dried fish. He once visited Meng Lingxiu; Lingxiu had previously suffered from cautery sores, and the scabs had fallen upon the bed, whereupon Liu Yong took and ate them. Lingxiu was greatly alarmed. Liu Yong replied, &quot;It is my nature to love this.&quot; Lingxiu then stripped away all remaining scabs from his body to provide for Liu Yong. After Liu Yong departed, Lingxiu wrote to He Xu, saying, &quot;Liu Yong just looked at me and devoured me, until my whole body bled.&quot; In Nankang Commandery, some two hundred officials, regardless of whether they were guilty or innocent, were whipped in rotation so that the resulting scabs might constantly provide for his meals.
#: '''Original:''' （刘）邕所至嗜食疮痂，以为味似鳆鱼。尝诣孟灵休，灵休先患灸疮，疮痂落床上，因取食之。灵休大惊。答曰：“性之所嗜。”灵休疮痂未落者，悉褫取以饴邕。邕既去，灵休与何勖书曰：“刘邕向顾见啖，遂举体流血。”南康国吏二百许人，不问有罪无罪，递互与鞭，鞭疮痂常以给膳。
#: '''Source:''' [[:w:Book of Song|Book of Song]], Vol. 42 &quot;Biographies, 2: Liu Muzhi et al.&quot; (《宋书·卷四十二·列传第二·刘穆之等》)
# c. 454 CE: Liu Yong's Consumption of Scabs, ''Nan Shi''
#: '''English''': Liu Yong had a passion for eating scabs, believing the taste resembled abalone. He once visited Meng Lingxiu; Lingxiu had previously suffered from blistions caused by [[:w:Moxibustion|moxibustion]], and the scabs fell upon the bed, which Liu Yong took and ate. Lingxiu was greatly alarmed; he then stripped away all remaining scabs to provide for Liu Yong. After Liu Yong departed, Lingxiu wrote to He Xu, saying, &quot;Liu Yong just looked at me and devoured me, until my whole body bled.&quot; In Nankang Commandery, some two hundred officials, regardless of whether they were guilty or innocent, were whipped in rotation, and the scabs were constantly provided for his meals.
#: '''Original:''' （刘）邕性嗜食疮痂，以为味似鳆鱼。尝诣孟灵休，灵休先患灸疮，痂落在床，邕取食之。灵休大惊，痂未落者，悉褫取饴邕。邕去，灵休与何勖书曰：“刘邕向顾见啖，遂举体流血。”南康国吏二百许人，不问有罪无罪，递与鞭，疮痂常以给膳。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 15 &quot;Biographies, 5: Liu Muzhi et al.&quot; (《南史·卷十五·列传第五·刘穆之等》)
# 465 CE: Mutilation of Wang Yigong, ''Nan Shi''
#: '''English''': The former deposed Emperor (Liu Ziye) was maddened and lawless. Wang Yigong and Liu Yuanjing conspired to depose him; the deposed Emperor led the Yulin guards to their residences and slew them, along with their four sons. He cut and severed the limbs of Wang Yigong, split open his abdomen and stomach, and plucked out his eyes to soak them in honey, calling them &quot;Ghost-Eye [[:w:Zongzi|Zongzi]].&quot;
#: '''Original:''' 前废帝（刘子业）狂悖无道，（王）义恭、（柳）元景谋欲废立，废帝率羽林兵于第害之，并其四子。断析义恭支体，分裂腹胃，挑取眼睛以蜜渍之，以为鬼目粽。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 13 &quot;Biographies, 3: Imperial Clan and Various Princes&quot; (《南史·卷十三·列传第三·宋宗室及诸王上》)
## 465 CE: Mutilation of Wang Yigong, ''Zizhi Tongjian''
##: '''English''': The Emperor (the former deposed Emperor of the Southern Song, Liu Ziye) personally led the Yulin guards to attack Wang Yigong and slew him, along with his four sons. He severed the limbs of Wang Yigong, split open his intestines and stomach, plucked out his eyes, and soaked them in honey, calling them &quot;Ghost-Eye Zongzi.&quot;
##: '''Original:''' 帝（南朝宋前废帝刘子业）自帅羽林兵讨（王）义恭，杀之，并其四子。断绝义恭支体，分裂肠胃，挑取眼睛，以蜜渍之，谓之“鬼目粽”。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 130 (《资治通鉴》卷130)
# 498 CE: Cannibalism of Huang Yaoqi (1), ''Book of Southern Qi''
#: '''English''': The barbarian forces pursued and captured Huang Yaoqi; Wang Su recruited men to carve up and eat his flesh.
#: '''Original:''' 虏追军获（黄）瑶起，王肃募人脔食其肉。
#: '''Source:''' [[:w:Book of Southern Qi|Book of Southern Qi]], Vol. 57 &quot;Biographies, 38: Wei Barbarians&quot; (《南齐书·卷五十七·列传第三十八·魏虏》)
## 498 CE: Cannibalism of Huang Yaoqi, ''Nan Shi''
##: '''English''': Wang Chen's brothers, Su and Bing, both fled to Wei; later they captured Huang Yaoqi, carved him up, and ate him.
##: '''Original:''' （王）琛弟肃、秉并奔魏，后得黄瑶起脔食之。
##: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 23 &quot;Biographies, 13: Wang Dan et al.&quot; (《南史·卷二十三·列传第十三·王诞等》)
## 498 CE: Cannibalism of Huang Yaoqi, ''Zizhi Tongjian''
##: '''English''': Huang Yaoqi was captured by Wei; the Lord of Wei bestowed him upon Wang Su, who carved him up and ate him.
##: '''Original:''' （黄）瑶起为魏所获，魏主以赐王肃，肃脔而食之。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 141 (《资治通鉴》卷141)
# 499 CE: Siege of Maquan City, ''Book of Southern Qi''
#: '''English''': In the first year of Yongyuan, Chen Xianda supervised General Cui Huijing and forty thousand troops to besiege Maquan City in Nanxiang, three hundred li from Xiangyang, attacking for forty days. The barbarians' food was exhausted; they ate the flesh of dead men and tree bark.
#: '''Original:''' 永元元年，（陈）显达督平北将军崔慧景众军四万，围南乡堺马圈城，去襄阳三百里，攻之四十日。虏食尽，啖死人肉及树皮。
#: '''Source:''' [[:w:Book of Southern Qi|Book of Southern Qi]], Vol. 26 &quot;Biographies, 7: Wang Jingze, Chen Xianda&quot; (《南齐书·卷二十六·列传第七·王敬则 陈显达》)
## 499 CE: Siege of Maquan City, ''Nan Shi''
##: '''English''': In the first year of Yongyuan, Chen Xianda supervised General Cui Huijing and forty thousand troops to besiege Maquan City in Nanxiang, three hundred li from Xiangyang. They attacked for forty days; the Wei army's food was exhausted, and they ate the flesh of dead men and tree bark.
##: '''Original:''' 永元元年，（陈）显达督平北将军崔慧景众军四万，围南乡界马圈城，去襄阳三百里。攻之四十日，魏军食尽，啖死人肉及树皮。
##: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 45 &quot;Biographies, 35: Wang Jingze et al.&quot; (《南史·卷四十五·列传第三十五·王敬则等》)
## 499 CE: Siege of Maquan City, ''Zizhi Tongjian''
##: '''English''': Chen Xianda fought Wei Yuanying and repeatedly defeated him. He sieged Maquan City for forty days; the food within the city was exhausted, and they ate the flesh of dead men and tree bark.
##: '''Original:''' 陈显达与魏元英战，屡破之。攻马圈城四十日，城中食尽，啖死人肉及树皮。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 142 (《资治通鉴》卷142)
# 502 CE: Aftermath of Sun Wenming's Rebellion, Nan Shi
#: '''English''': At that time, the remnants of the Eastern Tyrant, including Sun Wenming and others, rebelled. Zhang Hongce jumped over a wall to hide in the dragon stables, where he encountered rebels and was thereupon slain. The government army captured Sun Wenming and executed him in the East Market; the kinsmen of the Zhang family carved him up and ate him.
#: '''Original:''' 时东昏余党孙文明等……作乱，……（张）弘策踰垣匿于龙厩，遇贼见害。……官军捕文明斩于东市，张氏亲属脔食之。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 56 &quot;Biographies, 46: Zhang Hongce et al.&quot; (《南史·卷五十六·列传第四十六·张弘策等》)
#  503年: 成都城中食尽，升米三千，人相食。（《资治通鉴》卷145）
#   约525年: 大将军萧宝夤西讨，德广为行台郎，募众而征，战捷，乃手刃仇人，啖其肝肺。（《北史·卷一百·序传第八十八》㉕*）
#   525年: 山胡刘蠡升自云圣术，胡人信之，咸相影附，旬日之间，逆徒还振。……先是官粟贷民。未及收聚，仍值寇乱。至是（汾州）城民大饥，人相食。贼知仓库空虚，攻围日甚，死者十三四。（裴）良以饥窘，因与城人奔赴西河。（《魏书·卷六十九·列传第五十七·崔休等》㉕*）
#  529年: 于是（元颢）斩（杨）昱下统帅三十七人，皆令蜀兵刳腹取心食之。（《北史·卷四十一·列传第二十九·杨播等》㉕*）&lt;p&gt;于是（元颢）斩（杨）昱下统帅三十七人，皆令蜀兵刳腹取心食之。（《魏书·卷五十八·列传第四十六·杨播》㉕）&lt;/p&gt;&lt;p&gt;于是（元颢）斩（杨）昱所部统帅三十七人，皆刳心而食之。 （《资治通鉴》卷153）&lt;/p&gt;
#   约532年：（北方）于时年凶，人多相食，昕勤恤人隐，多所全济。（《北史·卷二十四·列传第十二·崔逞等》㉕*）
#   约533年: 中大通四年，（梁武帝萧衍）特封（萧正德）临贺郡王。后为丹阳尹，坐所部多劫盗，复为有司所奏，去职。出为南兖州，在任苛刻，人不堪命。广陵沃壤，遂为之荒，至人相食啖。（《南史·卷五十一·列传第四十一·梁宗室上》㉕*）
#  536年: 是岁，关中大饥，人相食，死者十七八。（《北史·卷五·魏本纪第五》㉕*）&lt;p&gt; （西）魏关中大饥，人相食，死者什七八。  （《资治通鉴》卷157）&lt;/p&gt;
#  548年: 景食石头常平仓既尽，便掠居人，尔后米一升七八万钱，人相食，有食其子者。又筑土山，不限贵贱，昼夜不息，乱加殴棰，疲羸者因杀以填山，号哭之声动天地。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;石头常平诸仓既尽，（侯景）军中乏食；乃纵士卒掠夺民米及金帛子女。是后米一升直七八万钱，人相食，饿死者什五六。 （《资治通鉴》卷161）&lt;/p&gt;
#   548年: 鄱阳世子嗣、永安侯确、羊鸦仁、李迁仕、樊文皎率众度淮，攻破贼（侯景）东府城前栅，遂营于青溪水东。（侯）景遣其仪同宋子仙缘水西立栅以相拒。景食稍尽，人相食者十五六。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;景遣其仪同宋子仙顿南平王第，缘水西立栅相拒。景食稍尽，至是米斛数十万，人相食者十五六。（《梁书·卷五十六·列传第五十·侯景》㉕）&lt;/p&gt;
#   549年， [[:w:梁武帝|梁武帝]]太清三年：贼（侯景）之始至，（建邺）城中才得固守，平荡之事，期望援军。既而中外断绝，……军人屠马于殿省间鬻之，杂以人肉，食者必病。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;（萧）衍城内大饥，人相食，米一斗八十万，皆以人肉杂牛马而卖之。（《魏书·卷九十八·列传第八十六·岛夷萧道成等》㉕）&lt;/p&gt;&lt;p&gt;（梁）军人屠马于殿省间，杂以人肉，食者必病。 （《资治通鉴》卷162）&lt;/p&gt;
#  549年: 自（侯）景作乱，（建康）道路断绝，数月之间，人至相食，犹不免饿死，存者百无一二。贵戚、豪族皆自出采稆，填委沟壑，不可胜纪。  （《资治通鉴》卷162）
#   549年，梁太清三年：是月（七月），九江大饥，人相食十四五。（《梁书·卷四·本纪第四·简文帝》㉕*）&lt;p&gt;九江大饥，人相食者十四五。（《南史·卷八·梁本纪下第八》㉕）&lt;/p&gt;&lt;p&gt;是年，帝为侯景所幽，崩。七月，九江大饥，人相食十四五。（《隋书·卷二十一·志第十六·天文下》㉕）&lt;/p&gt;
#  550: 值梁室丧乱，（姚察）于金陵随二亲还乡里。时东土兵荒，人饥相食，告籴无处，察家口既多，并采野蔬自给。（《陈书· 卷二十七·列传第二十一·江总 姚察》㉕*）&lt;p&gt;自晋氏度江，三吴最为富庶，贡赋商旅，皆出其地。及侯景之乱，掠金帛既尽，乃掠人而食之，或卖于北境，遗民殆尽矣。  （《资治通鉴》卷163）&lt;/p&gt;
#   550年，梁大宝元年：自春迄夏，大饥，人相食，京师尤甚。（《梁书·卷四·本纪第四·简文帝》㉕*）&lt;p&gt;自春迄夏大旱，人相食，都下尤甚。（《南史·卷八·梁本纪下第八》㉕）&lt;/p&gt;
#   552年:（侯）景不能制，乃与腹心数十人单舸走，推堕二子于水，自沪渎入海。至壶豆洲，前太子舍人羊鲲杀之，送尸于王僧辩，传首西台，曝尸于建康市。百姓争取屠脍啖食，焚骨扬灰。（《梁书·卷五十六·列传第五十·侯景》㉕*）&lt;p&gt;及（侯）景死，僧辩截其二手送齐文宣，传首江陵，果以盐五斗置腹中，送于建康，暴之于市。百姓争取屠脍羹食皆尽，并溧阳主亦预食例。景焚骨扬灰，曾罹其祸者，乃以灰和酒饮之。（《南史·卷八十·列传第七十·贼臣》㉕）&lt;/p&gt;&lt;p&gt;既斩侯景，烹尸于建业市，百姓食之，至于肉尽龁骨，传首荆州，悬于都街。（《北齐书· 卷四十五·列传第三十七·文苑》㉕）&lt;/p&gt;&lt;p&gt;僧辩传（侯景）首江陵，截其手，使谢葳蕤送于齐；暴景尸于市，士民争取食之，并骨皆尽；溧阳公主亦预食焉。 （《资治通鉴》卷164）&lt;/p&gt;
#   552年: 王伟，陈留人。少有才学，景之表、启、书、檄，皆其所制。景既得志，规摹篡夺，皆伟之谋。及囚送江陵，烹于市，百姓有遭其毒者，并割炙食之。（《梁书·卷五十六·列传第五十·侯景》㉕*）
#  553年: （萧）圆照更无所言，唯云计误。并命绝食于狱，齿臂啖之，十三日死，天下闻而悲之。（《南史·卷五十三·列传第四十三·梁武帝诸子》㉕*）&lt;p&gt;上（梁元帝萧绎）并命（萧圆正）绝食于狱，至啮臂啖之，十三日而死，远近闻而悲之。 （《资治通鉴》卷165）&lt;/p&gt;
#  《南史》毗骞：“国法刑人，并于王前啖其肉。”“国内不受估客，往者亦杀而食之。”
#  554年: 五年春正月癸丑，帝（北齐文宣帝高洋）讨山胡大破之。男子十二已上皆斩，女子及幼弱以赏军。遂平石楼。石楼绝险，自魏代所不能至。于是远近山胡，莫不慑伏。是役也，有都督战伤，其什长路晖礼不能救，帝命刳其五藏，使九人分食之，肉及秽恶皆尽。自是始行威虐。（《北史·卷七·齐本纪中第七》㉕*）&lt;p&gt;有都督战伤，其什长路晖礼不能救，帝（北齐文宣帝高洋）命刳其五藏，令九人食之，肉及秽恶皆尽。（《资治通鉴》卷165）&lt;/p&gt;
#   555年: 众推（慕容）俨，遂遣镇郢城。……（侯）瑱、（任）约又并力围城。唯煮槐楮叶并纻根、水荭、葛、艾等及靴、皮带、筋角等食之。人死，即火别分食，唯留骸骨。俨犹信赏必罚，分甘同苦。自正月至六月，人无异志。（《北史·卷五十三·列传第四十一·万俟普等》㉕*）
#   约555年-560年: 自（天保）六年之后，帝（北齐文宣帝高洋）遂以功业自矜，恣行酷暴，昏狂酗醟，任情喜怒。为大镬、长锯、剉碓之属，并陈于庭，意有不快，则手自屠裂，或命左右脔啖，以逞其意。（《隋书·卷二十五·志第二十·刑法》㉕*）
#  流求国，居海岛，当建安郡东。水行五日而至。……国人好相攻击，……两军相当，勇者三五人出前跳噪，交言相骂，因相击射。如其不胜，一军皆走，遣人致谢，即共和解。收取斗死者聚食之，仍以髑髅将向王所，王则赐之以冠，便为队帅。……其南境风俗少异，人有死者，邑里共食之。（《北史·卷九十四·列传第八十二·高丽等》㉕*）&lt;p&gt;流求国，……南境风俗少异，人有死者，邑里共食之。（《隋书·卷八十一·列传第四十六·东夷》㉕）&lt;/p&gt;
#   獠者，盖南蛮之别种，自汉中达于邛、笮，川洞之间，所在皆有。……性同禽兽，至于忿怒，父子不相避，唯手有兵刃者先杀之。……若报怨相攻击，必杀而食之；（《北史·卷九十五·列传第八十三·蛮 獠 等》㉕*）
#   顿逊之外，大海洲中，又有毗骞国，去扶南八千里。……国法刑罪人，并于王前啖其肉。国内不受估客，有往者亦杀而啖之，是以商旅不敢至。（《梁书·卷五十四·列传第四十八·诸夷》㉕*）&lt;p&gt;又有毗骞国，去扶南八千里。……国法刑人，并于王前啖其肉。国内不受估客，有往者亦杀而啖之，是以商旅不敢至。（《南史· 卷七十八·列传第六十八·夷貊上》㉕）&lt;/p&gt;

==隋==
#   590年: 时江南州县又论言欲徙之入关，远近惊骇。饶州吴世华起兵为乱，生脔县令，啖其肉。（《北史·卷六十三·列传第五十一·周惠达等》㉕*）
#   隋文帝开皇年间（581-600年）：（杨武通）与周法尚讨嘉州叛獠，……贼知其孤军无援，倾部落而至。武通转斗数百里，为贼所拒，四面路绝。武通轻骑挑战，坠马，为贼所执，杀而啖之。（《北史·卷七十三·列传第六十一·梁士彦等》㉕*）&lt;p&gt;（杨）武通轻骑接战，坠马，为贼所执，杀而啖之。（《隋书·卷五十三·列传第十八·达奚长儒》㉕）&lt;/p&gt;
#   隋文帝开皇年间（581-600年）：郡中士女，号哭于路，诸郡惊骇，各奏其（王文同）事。帝闻大怒，遣使者违奚善意驰锁之（王文同），斩于河间，以谢百姓。仇人剖其棺，脔其肉啖之，斯须咸尽。（《北史·卷八十七·列传第七十五·酷吏》㉕*）&lt;p&gt;郡中士女号哭于路，诸郡惊骇，各奏其（王文同）事。帝闻而大怒，遣使者达奚善意驰锁之，斩于河间，以谢百姓，仇人剖其棺，脔其肉而啖之，斯须咸尽。（《隋书·卷七十四·列传第三十九·酷吏》㉕）&lt;/p&gt;
#   隋炀帝时代（604年-618年在位）中期：六军不息，百役繁兴；行者不归，居者失业；人饥相食，邑落为墟，上弗之恤也。（《北史·卷十二·隋本纪下第十二》㉕*）&lt;p&gt;六军不息，百役繁兴，行者不归，居者失业。人饥相食，邑落为墟，上不之恤也。（《隋书·卷四·帝纪第四·炀帝下》㉕）&lt;/p&gt;
#   613年: 及杨玄感反，帝（隋炀帝杨广）诛之，罪及九族。其尤重者，行轘裂枭首之刑。或磔而射之。命公卿已下，脔啖其肉。（《隋书·卷二十五·志第二十·刑法》㉕*）
#  614年:明年，（隋炀）帝复东征，高丽请和，遂送（斛斯）政。锁至京师以告庙，左翊卫大将军宇文述请变常法行刑，帝许之。以出金光门，缚之于柱，公卿百僚，并亲击射。脔其肉，多有啖者，然后烹焚，扬其骨灰。（《北史·卷四十九·列传第三十七·朱瑞等》㉕*）&lt;p&gt;（隋炀）帝复东征，高丽请降，求执送（斛斯）政。帝许之，遂锁政而还。至京师，以政告庙，左翊卫大将军字文述奏曰：“斛斯政之罪，天地所不容，人神所同忿。若同常刑，贼臣逆子何以惩肃？请变常法。”帝许之。于是将政出金光门，缚政于柱，公卿百僚并亲击射，脔割其肉，多有啖者。啖后烹煮，收其余骨，焚而扬之。（《隋书·卷七十·列传第三十五·杨玄感》㉕）&lt;/p&gt;&lt;p&gt;十一月，丙申，杀斛斯政于金光门外，如杨积善之法，仍烹其肉，使百官啖之，佞者或啖之至饱，收其馀骨，焚而扬之。 （《资治通鉴》卷182）&lt;/p&gt;
#   隋炀帝时代（604年-618年在位）后期：民外为盗贼所掠，内为郡县所赋，生计无遗；加之饥馑无食，民始采树皮叶，或捣穢为末，或煮土而食之，诸物皆尽，乃自相食。而官食犹充牣，吏皆畏法，莫敢振救。 （《资治通鉴》卷183）&lt;p&gt;相聚雚蒲，猬毛而起。大则跨州连郡，称帝称王；小则千百为群，攻城剽邑。流血成川泽，死人如乱麻；炊者不及析骸，食者不遑易子。（《北史·卷十二·隋本纪下第十二》㉕*）&lt;/p&gt;&lt;p&gt;俄而玄感肇黎阳之乱，匈奴有雁门之围，天子方弃中土，远之扬越。奸宄乘衅，强弱相陵，关梁闭而不通，皇舆往而不反。加之以师旅，因之以饥馑，流离道路，转死沟壑，十八九焉。于是相聚萑蒲，蝟毛而起，大则跨州连郡，称帝称王，小则千百为群，攻城剽邑，流血成川泽，死人如乱麻，炊者不及析骸，食者不遑易子。（《隋书·卷四·帝纪第四·炀帝下》㉕）&lt;/p&gt;&lt;p&gt;自燕赵跨于齐韩，江淮入于襄邓，东周洛邑之地，西秦陇山之右，僭伪交侵，盗贼充斥。宫观鞠为茂草，乡亭绝其烟火，人相啖食，十而四五。（《隋书·卷二十四·志第十九·食货》㉕）&lt;/p&gt;&lt;p&gt;是时百姓废业，屯集城堡，无以自给。然所在仓库，犹大充爨，吏皆惧法，莫肯赈救，由是益困。初皆剥树皮以食之，渐及于叶，皮叶皆尽，乃煮土或捣稿为末而食之。其后人乃相食。（《隋书·卷二十四·志第十九·食货》㉕）&lt;/p&gt;
#  616: 吏立木于市，悬其（张金称）头，张其手足，令仇家割食之；未死间，歌讴不辍。（《资治通鉴》卷183）
#  617年，大业十三年四月：（薛仁杲）所至多杀人，纳其妻妾。获庾信子立，怒其不降，磔于猛火之上，渐割以啖军士。（《旧唐书·卷五十五·列传第五·薛举等》㉕*）&lt;p&gt;（薛仁杲）尝得庾信子立，怒其不降，砾之火，渐割以啖士。（《新唐书·卷八十六·列传第十一 薛李二刘高徐》㉕）&lt;/p&gt;&lt;p&gt;（薛仁杲）尝获庾信子立，怒其不降，磔于火上，稍割以啖军士。”（《资治通鉴》卷183）&lt;/p&gt;
#  618: :（屈突）通引兵南遁，置（尧）君素领河东通守。……后颇得江都倾覆消息，又粮尽，男女相食，众心离骇。（《北史·卷八十五·列传第七十三·节义》㉕*）&lt;p&gt;时百姓苦隋日久，及逢义举，人有息肩之望。然君素善于统领，下不能叛。岁余，颇得外生口，城中微知江都倾覆。又粮食乏绝，人不聊生，男女相食，众心离骇。（《隋书·卷七十一·列传第三十六·诚节》㉕）&lt;/p&gt;&lt;p&gt;隋将尧君素守河东，上遣吕绍宗、韦义节、独孤怀恩相继攻之，俱不下。……久之，仓粟尽，人相食；（《资治通鉴》卷184）&lt;/p&gt;
#  618: （李轨）征兵筑台以候玉女，多所糜费，百姓患之。又属年饥，人相食，轨倾家赈之，私家罄尽，不能周遍。（谢统师等）乃诟珍曰：“百姓饿者自是弱人，勇壮之士终不肯困，国家仓粟须备不虞，岂可散之以供小弱？仆射苟悦人情，殊非国计。”轨以为然，由是士庶怨愤，多欲叛之。（《旧唐书·卷五十五·列传第五 薛举等》㉕*）&lt;p&gt;有胡巫妄曰：“上帝将遣玉女从天来。”（李轨）遂召兵筑台以候女，多所糜损。属荐饥，人相食，轨毁家赀赈之，不能给，议发仓粟，曹珍亦劝之。谢统师等故隋官，内不附，每引结群胡排其用事臣，因是欲离沮其众，乃廷诘珍曰：“百姓饥死皆弱不足事者，壮勇士终不肯困。且储禀以备不虞，岂宜妄散惠孱小乎？仆射苟附下，非国计。”轨曰：“善。”乃闭粟。下益怨，多欲叛去。（《新唐书·卷八十六·列传第十一·薛李二刘高徐》㉕） &lt;/p&gt;&lt;p&gt;有胡巫谓（李）轨曰：“上帝当遣玉女自天而降。”轨信之，发民筑台以候玉女，劳费甚广。河右饥，人相食，轨倾家财以赈之；不足，欲发仓粟，召群臣议之。曹珍等皆曰：“国以民为本，岂可爱仓粟而坐视其死乎！”谢统师等皆故隋官，心终不服，密与群胡为党，排轨故人，乃诟珍曰：“百姓饿者自是羸弱，勇壮之士终不至此。国家仓粟以备不虞，岂可散之以饲羸弱！仆射苟悦人情，不为国计，非忠臣也。”轨以为然，由是士民离怨。 （《资治通鉴》卷186）&lt;/p&gt;
#  619年：（朱）粲所克州县，皆发其藏粟以充食，迁徙无常，去辄焚余赀，毁城郭，又不务稼穑，以劫掠为业。于是百姓大馁，死者如积，人多相食。军中罄竭，无所虏掠，乃取婴儿蒸而啖之，因令军士曰：“食之美者，宁过于人肉乎！但令他国有人，我何所虑？”即勒所部，有略得妇人小儿皆烹之，分给军士，乃税诸城堡，取小弱男女以益兵粮。隋著作佐郎陆从典、通事舍人颜愍楚因谴左迁，并在南阳，粲悉引之为宾客，后遭饥馁，合家为贼所啖。（《旧唐书·卷五十六·列传第六·萧铣等》㉕*）&lt;p&gt;粲所克州县皆发藏粟以食，迁徙无常，去辄燔廥聚，毁城郭，不务稼穑，专以劫为资。于是人大馁，死者系路，其军亦匮，乃掠小儿烝食之。戒其徒曰：“味之珍宁有加人者？弟使佗国有人，我恤无储哉！”勒所部略妇人孺儿分烹之，又税诸城细弱以益粮。隋著作佐郎陆从典、通事舍人颜愍楚谪南阳，粲初引为宾客，后尽食两家。俄而诸城惧，皆逃散。（《新唐书·卷八十七·列传第十二·萧辅沈李梁》㉕）&lt;/p&gt;&lt;p&gt;朱粲有众二十万，剽掠汉、淮之间，迁徙无常，攻破州县，食其积粟未尽，复他适，将去，悉焚其余资；又不务稼穑，民馁死者如积。粲无可复掠，军中乏食，乃教士卒烹妇人、婴儿啖之，曰：“肉之美者无过于人，但使他国有人，何忧于馁！”隋著作佐郎陆从典、通事舍人颜愍楚，谪官在南阳，粲初引为宾客，其后无食，阖家皆为所啖。愍楚，之推之子也。又税诸城堡细弱以供军食，诸城堡相帅叛之。”（《资治通鉴》）&lt;/p&gt;&lt;p&gt;“隋末荒亂，狂賊[[:w:朱粲|朱粲]]起於襄、鄧間，歲飢，米斛萬錢，亦無得處，人民相食。粲乃驅男女小大仰一大銅鐘，可二百石，煮人肉以矮賊。生靈殲於此矣。”，朱粲竟說：“食之美者，寧過於人肉乎！”（唐·[[:w:張鷟|張鷟]]《朝野僉載》）&lt;/p&gt;
#  619年: （段）确醉，戏（朱）粲曰：“君脍人多矣，若为味？”粲曰：“啖嗜酒人，正似糟豚。”确悸，骂曰：“狂贼，归朝乃一奴耳，复得噬人乎？”粲惧，收确于坐，并从者数十悉饔之，以飨左右。遂屠菊潭，奔王世充，署龙骧大将军。东都平，斩洛水上。士庶竞掷瓦砾击其尸，须臾若冢。（《新唐书·卷八十七·列传第十二·萧辅沈李梁》㉕*）&lt;p&gt;（段确）乘醉侮（朱）粲曰：“闻卿好啖人，人作何味？”粲曰：“啖醉人正如糟藏彘肉。”确怒，骂曰：“狂贼入朝，为一头奴耳，复得啖人乎！”粲于座收确及从者数十人，悉烹之，以啖左右。（《资治通鉴》卷187）&lt;/p&gt;
#  隋末的[[:w:诸葛昂|诸葛昂]]與[[:w:高瓒|高瓒]]嗜食人肉。高瓒將双胞胎小孩杀掉，頭顱、手和腳分別裝在盤子裏，做成“双子宴”，與诸葛昂一起享用；诸葛昂则把自己的爱妾蒸熟，擺成盤腿打坐的姿勢，臉上重新塗好脂粉，諸葛昂親手撕她大腿上的肉請高瓒吃。（《[[:w:唐人说荟|唐人说荟]]》卷五，引张骞《耳目记》）

==唐==
安史之乱期间，张巡固守城池，城中人相食，张巡杀妾以飨将士，对于张巡以食人为代价的守土之功是否应该奖励，出现了一次伦理学的辩论，历代不息，《柏杨白话版资治通鉴》收集了若干历史上争论的意见。

黄巢之乱的时候，几支反叛军队成规模地常规性地以人为食，黄巢军“掠人为粮，生投于碓硙，并骨食之，号给粮之处曰‘舂磨寨’”，秦宗权军“啖人为储，军士四出，则盐尸而从”，李罕之军“不耕稼，专以剽掠为资，啖人为粮”。真是惨烈之甚。

唐朝陈藏器写的《本草拾遗》写人肉可以治病，这应该不是他的发明，而只是民间认知的一种总结，可能只是太多不得已的饥荒食人造成一种认知扭曲，但又反过来理性化了食人，到宋朝的时候，割肉疗亲开始出现。

#   621年，[[:w:唐高祖|唐高祖]]武德四年：（王）世充屯兵不散，仓粟日尽，城中人相食。或握土置瓮中，用水淘汰，沙石沉下，取其上浮泥，投以米屑，作饼饵而食之，人皆体肿而脚弱，枕倚于道路。其尚书郎卢君业、郭子高等皆死于沟壑。（《旧唐书·卷五十四·列传第四 王世充 窦建德》㉕*）&lt;p&gt;王（李世民）傅城，堑而守之。（王）世充粮且尽，人相食，至以水汨泥去砾，取浮土糅米屑为饼。民病肿股弱，相藉倚道上，其尚书郎卢君业、郭子高等皆饿死。御史大夫郑颋丐为浮屠，世充恶其言，杀之。（《新唐书·卷八十五·列传第十 王窦》㉕）&lt;/p&gt;
#621年: （单雄信）临将就戮，（李世）勣对之号恸，割股肉以啖之，曰：“生死永诀，此肉同归于土矣。”（《旧唐书·卷六十七·列传第十七·李靖等》㉕*）&lt;p&gt;（李世勣）乃割股肉以啖（单）雄信，曰：“使此肉随兄为土，庶几犹不负昔誓也！”（《资治通鉴》卷189）&lt;/p&gt;
#   627年: （王）君操密袖白刃刺杀之（杀父仇人李君则），刳腹取其心肝，啖食立尽，诣刺史具自陈告。（《旧唐书·卷一百八十八·列传第一百三十八·孝友》㉕*）
#   643年，[[:w:唐太宗|唐太宗]]贞观十七年: 贞观末，（刘兰）以谋反腰斩。右骁卫大将军丘行恭探其心肝而食之，太宗闻而召行恭让之曰：“典刑自有常科，何至于此！必若食逆者心肝而为忠孝，则刘兰之心为太子诸王所食，岂至卿邪？”行恭无以答。（《旧唐书·卷六十九·列传第十九·侯君集等》㉕*）&lt;p&gt;鄠尉[[:w:游文芝|游文芝]]告代州都督[[:w:劉蘭成|劉蘭成]]谋反，戊申，兰成坐[[:w:腰斩|腰斩]]。右武候将军[[:w:丘行恭|丘行恭]]，探兰成心肝食之。上（唐太宗）闻而让之曰：兰成谋反，国有常刑，何至如此！若以为忠孝，则太子诸王先食之矣，岂至卿耶？行恭惭而拜谢。（《资治通鉴》卷196）&lt;/p&gt;
#   约650年：周智寿者，雍州同官人。其父永徽初被族人安吉所害。智寿及弟智爽乃候安吉于途，击杀之。兄弟相率归罪于县，争为谋首，官司经数年不能决。乡人或证智爽先谋，竟伏诛。临刑神色自若，顾谓市人曰：“父仇已报，死亦何恨！”智寿顿绝衢路，流血遍体。又收智爽尸，舐取智爽血，食之皆尽，见者莫不伤焉。（《旧唐书·卷一百八十八·列传第一百三十八·孝友》㉕*）
#   662年: （郑）仁泰选骑万四千卷甲驰，绝大漠，至仙萼河，不见虏，粮尽还。人饥相食，比入塞，余兵才二十之一。（《新唐书·卷一百一十一·列传第三十六·郭二张三王苏薛程唐》㉕*)&lt;p&gt;（郑）仁泰将轻骑万四千，倍道赴之，遂逾大碛，至仙萼河，不见虏，粮尽而还。值大雪，士卒饥冻，弃捐甲兵，杀马食之，马尽，人自相食，比入塞，馀兵才八百人。（《资治通鉴》卷200）&lt;/p&gt;
#   682年，[[:w:唐高宗|唐高宗]]永淳元年：关中先水后早蝗，继以疾疫，米斗四百，两京间死者相枕于路，人相食。”（《资治通鉴》卷203）&lt;p&gt;六月，关中初雨，麦苗涝损，后旱，京兆、岐、陇螟蝗食苗并尽，加以民多疫疠，死者枕藉于路，诏所在官司埋瘗。京师人相食，寇盗纵横。（《旧唐书·卷五本纪第五·高宗下》㉕*）&lt;/p&gt;&lt;p&gt;永淳中，为雍州长史。时关中大饥，人相食，盗贼纵横。（《旧唐书·卷七十五·列传第二十五·苏世长等》㉕）&lt;/p&gt;&lt;p&gt;是月，大蝗，人相食。（《新唐书·卷三·本纪第三·高宗》㉕）&lt;/p&gt;&lt;p&gt;永淳元年，关中及山南州二十六饥，京师人相食。（《新唐书·卷三十五·志第二十五》㉕）&lt;/p&gt;&lt;p&gt;（良嗣）徙雍州。时关内饥，人相食，良嗣政上严，每盗发，三日内必擒，号称神明。（《新唐书·卷一百三·列传第二十八·苏世长等》㉕）&lt;/p&gt;
#   约684年: 王友贞，怀州河内人也。父知敬，则天时麟台少监，以工书知名。友贞弱冠时，母病笃，医言唯啖人肉乃差。友贞独念无可求治，乃割股肉以饴亲，母病寻差。则天闻之，令就其家验问，特加旌表。（《旧唐书·卷一百九十二·列传第一百四十二·隐逸》㉕*）
#   [[:w:武則天|武則天]]時期，杭州臨安縣尉薛震好吃人肉，“有債主及奴詣臨安，于客舍，遂飲之醉。殺而臠之，以水銀和煎，并骨消盡。后又欲食其婦，婦覺而遁。縣令詰得其情，申州，錄事奏，奉敕杖殺之。”（《[[:w:朝野僉載|朝野僉載]]》）
#   武則天時期，“周岭南首陳元光設客，令一袍褲行酒。光怒，令曳出，遂殺之。須臾爛煮，以食諸客。后呈其二手，客懼，攫喉而吐。”（出《摭言》。明抄本作出《朝野僉載》）
#   697年: 丁卯，（李）昭德、（来）俊臣同弃市，时人无不痛昭德而快俊臣。仇家争啖俊臣之肉，斯须而尽，抉眼剥面，披腹出心，腾蹋成泥。（《资治通鉴》卷206）
#   张鷟《[[s:朝野僉載_(四庫全書本)/卷2|朝野佥载]]》卷二：“后诛易之昌宗等，百姓脔割其肉，肥白如猪肪，煎炙而食。”
#   唐玄宗開元中葉人[[:w:陳藏器|陳藏器]]（713年－741年）《[[:w:本草拾遺|本草拾遺]]》寫吃人肉可以治病。
#   739年: 内给事牛仙童使幽州，受张守珪厚赂。玄宗怒，命思勖杀之。思勖缚架之数日，乃探取其心，截去手足，割肉而啖之，其残酷如此。（《旧唐书·卷一百八十四·列传第一百三十四·宦官》㉕*）&lt;p&gt; 内给事牛仙童纳张守珪赂，诏付思勖杀之。思勖缚于格，箠惨不可胜，乃探心，截手足，剔肉以食，肉尽乃得死。（《新唐书·卷二百七·列传第一百三十二·宦者上》㉕）&lt;/p&gt;&lt;p&gt;739年: 上（唐玄宗李隆基）怒，甲戌，命杨思勖杖杀之（牛仙童）。思勖缚格，杖之数百，刳取其心，割其肉啖之。（《资治通鉴》卷214）&lt;/p&gt;
#   757年: （鲁）炅城中食尽，煮牛皮筋角而食之，米斗至四五十千，有价无米，鼠一头至四百文，饿死者相枕藉。……炅在围中一年，救兵不至，昼夜苦战，人相食。（《旧唐书·卷一百一十四·列传第六十四·鲁炅等》㉕*）&lt;p&gt;（鲁）炅被围凡一年，昼夜战，人至相食，卒无救。（《新唐书·卷一百四十七·列传第七十二·三王鲁辛冯三李曲二卢》㉕）&lt;/p&gt;
#   757年: 尹子奇攻围（睢阳）既久，城中粮尽，易子而食，析骸而爨，人心危恐，虑将有变。（张）巡乃出其妾，对三军杀之，以飨军士。曰：“诸公为国家戮力守城，一心无二，经年乏食，忠义不衰。巡不能自割肌肤，以啖将士，岂可惜此妇，坐视危迫。”将士皆泣下，不忍食，巡强令食之。乃括城中妇人；既尽，以男夫老小继之，所食人口二三万，人心终不离变。（《旧唐书·卷一百八十七下·列传第一百三十七·忠义下》㉕*）&lt;p&gt;（张）巡士多饿死，存者皆痍伤气乏。巡出爱妾曰：“诸君经年乏食，而忠义不少衰，吾恨不割肌以啖众，宁惜一妾而坐视士饥？”乃杀以大飨，坐者皆泣。巡强令食之，远亦杀奴僮以哺卒，至罗雀掘鼠，煮铠弩以食。……被围久，初杀马食，既尽，而及妇人老弱凡食三万口。人知将死，而莫有畔者。城破，遣民止四百而已。 （《新唐书·卷一百九十二·列传第一百一十七·忠义中》㉕） &lt;/p&gt;&lt;/p&gt;（张巡守睢阳，）茶纸既尽，遂食马；马尽，罗雀掘鼠；雀鼠又尽，巡出爱妾，杀以食士，远亦杀其奴；然后括城中妇人食之；既尽，继以男子老弱。人知必死，莫有叛者，所馀才四百人。 （《资治通鉴》卷220）&lt;/p&gt;
#   758年: 明年，改乾元元年，伪德州刺史王暕、贝州刺史宇文宽等皆归顺，河北诸军各以城守累月，贼使蔡希德、安太清急击，复陷于贼，虏之以归，脔食其肉。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕*）
#   759年: 二年正月，史思明自率范阳精卒复陷魏州，乃伪称燕王。王师虽众，军无统帅，进退无所承禀，自冬徂春，竟未破贼，但引漳水以灌其城，城中食尽，易子而食。（《旧唐书·卷一百二十·列传第七十·郭子仪等》㉕*）&lt;p&gt; （安）庆绪自十月被围至二月，城中人相食，米斗钱七万余，鼠一头直数千，马食隤墙麦鞬及马粪濯而饲之。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕）&lt;/p&gt;&lt;p&gt;（郭子仪军）连营进围相州，引漳水灌城，漫二时，不能破。城中粮尽，人相食。庆绪求救于史思明。（《新唐书·卷一百三十七·列传第六十二·郭子仪》㉕）&lt;/p&gt;&lt;p&gt; 乾元元年秋九月，帝诏郭子仪率九节度兵凡二十万讨庆绪，攻卫州，……王师围已固，筑浚城隍三周，决安阳水灌城。城中栈而处，粮尽，易口以食，米斗钱七万余，一鼠钱数千，屑松饲马，隤墙取麦秸，濯粪取刍，城中欲降不得。（《新唐书·卷二百二十五上·列传第一百五十上·逆臣上》㉕）&lt;/p&gt;
#   760年: 有纳赂于上言求官者，（吕）諲补之蓝田尉。五月，上言事泄笞死，以其肉令从官食之，諲坐贬太子宾客。（《旧唐书·卷一百八十五下·列传第一百三十五·良吏下》㉕*）
#   760年:   三品钱行浸久，属岁荒，米斗至七千钱，人相食。 （《资治通鉴》卷221）
#   760年: 时大雾，自四月雨至闰月末不止。米价翔贵，人相食，饿死者委骸于路。（《旧唐书·卷十·本纪第十·肃宗》㉕*）&lt;p&gt; 是时自四月初大雾大雨，至闰四月末方止。是月，逆贼史思明再陷东都，米价踊贵，斗至八百文，人相食，殍尸蔽地。（《旧唐书·卷三十六·志第十六·天文下》㉕） &lt;/p&gt;&lt;p&gt;乾元三年闰四月，大雾，大雨月余。是月，史思明再陷东都，京师米斗八百文，人相食，殍骸蔽地。（《旧唐书·卷三十七·志第十七·五行》㉕）&lt;/p&gt;
#   761年: 时洛阳四面数百里，人相食，州县为墟。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕*）&lt;p&gt; 朝义虚怀礼下，事皆决大臣，然无经略才。当此时，洛阳诸郡人相食，城邑榛墟，（《新唐书·卷二百二十五上·列传第一百五十上·逆臣上》㉕）&lt;/p&gt;
#  [[:w:唐代宗|唐代宗]]廣德元年（763年），江東大疫，“死者過半”，[[:w:獨孤及|獨孤及]]描述這次的災難：“辛丑歲（762年），大旱，三吳飢甚，人相食。明年大疫，死者十七八，城郭邑居為之空虛，而存者無食，亡者無棺殯悲哀之送。大抵雖其父母妻子也啖其肉，而棄其骸於田野，由是道路積骨相支撐枕藉者彌二千里，春秋以來不書。”（《吊道殣文》）&lt;p&gt;江、淮大饥，人相食。（《资治通鉴》卷222）&lt;/p&gt;
#   [[:w:白居易|白居易]]（772年－846年）寫《輕肥》一詩有“是歲江南旱，衢州人食人。”
#   [[:w:張茂昭|張茂昭]]為節鎮，頻吃人肉，及除統軍，到京。班中有人問曰：聞尚書在鎮好人肉，虛實？” 昭笑曰：“人肉腥而且肕，爭堪吃。”（《盧氏雜記》）
#   766年: 监军张志斌自陕入奏，（周）智光馆给礼慢，志斌责其不肃。智光大怒曰：“仆固怀恩岂有反状！皆由尔鼠辈作福作威，惧死不敢入朝。我本不反，今为尔作之。”因叱下斩之，脔其肉以饲从者。（《旧唐书·卷一百一十四·列传第六十四·鲁炅等》㉕*）&lt;p&gt;（周智光）叱下斩之（张志斌），脔食其肉。（《资治通鉴》卷224）&lt;/p&gt;
#   775年:承嗣既令（田）廷玠（或作田庭玠）守沧州，而（李）宝臣、朱滔兵攻击，欲兼其土宇。廷玠婴城固守，连年受敌，兵尽食竭，人易子而食，卒无叛者，卒能保全城守。（《旧唐书·卷一百四十一·列传第九十一·田承嗣等》㉕*）
#   796年: 军士又呼曰：“仓官刘叔何给纳有奸。”杀而食之。（《资治通鉴》卷235）
#   799年: 是日，汴州军乱，杀陆长源及节度判官孟叔度、丘颖，军人脔而食之。（《旧唐书·卷十三·本纪第十三·德宗下》㉕*）&lt;p&gt;兵士怨怒滋甚，乃执长源及叔度等脔而食之，斯须骨肉糜散。（《旧唐书·卷一百四十五·列传第九十五·刘玄佐等》㉕）&lt;/p&gt;&lt;p&gt;才八日，军乱，杀长源及叔度等，食其肉，放兵大掠。（《新唐书·卷一百五十一·列传第七十六·关董袁赵窦》㉕）&lt;/p&gt;&lt;p&gt;是日，军士作乱，杀（陆）长源、（孟）叔度，脔食之，立尽。（《资治通鉴》卷235）&lt;/p&gt;
#   803年: 盐夏节度判官崔文先权知盐州，为政苛刻。冬，闰十月，庚戌，部将李庭俊作乱，杀而脔食之。（《资治通鉴》卷236）
#   807年: 锜不自安，亦请入朝，乃拜锜左仆射。锜乃署判官王澹为留后。既而迁延发期，澹与中使频喻之，不悦，遂讽将士以给冬衣日杀澹而食之。监军使闻乱，遣衙将赵锜慰喻，又脔食之。（《旧唐书·卷一百一十二·列传第六十二·李暠等》㉕*）&lt;p&gt;会使者召锜，称疾，留后王澹为具行，锜怒，阴教士脔食之，即胁使者为众奏天子，幸得留。（《新唐书·卷一百八十一·列传第一百六·陈夷行等》㉕）&lt;/p&gt;&lt;p&gt;807: （李）锜严兵坐幄中，（王）澹与敕使入谒，有军士数百噪于庭曰：“王澹何人，擅主军务！”曳下，脔食之；大将赵琦出慰止，又脔食之（《资治通鉴》卷237）&lt;/p&gt;
#   817年: 蔡将有李端者，过溵河降重胤。其妻为贼束缚于树，脔食至死，将绝，犹呼其夫曰：“善事乌仆射。”（《旧唐书·卷一百六十一·列传第一百一十一·李光进等》㉕*）&lt;p&gt;李湍妻。湍，吴元济之军人也。元和中，淮南未平，湍心怀向顺，乃急渡溵河，东降乌重胤。其妻遂为贼束缚在树，脔而食之，至死，叫其夫曰：“善事乌仆射。”观者义之。至是，重胤以其事请列史册。十三年，宪宗下诏从之。（《旧唐书·卷一百九十四上·列传第一百四十四上·突厥上》㉕）&lt;/p&gt;&lt;p&gt;李湍妻某氏。湍籍吴元济军，元和中，自拔归鸟重胤，妻为贼缚而脔食之，将死，犹号湍曰：“善事鸟仆射！”观者叹泣。重胤请以其事属史官，诏可。（《新唐书·卷二百五·列传第一百三十·列女》㉕）&lt;/p&gt;
#   822年: （王）播至淮南，属岁旱俭，人相啖食，课最不充，设法掊敛，比屋嗟怨。（《旧唐书·卷一百六十四·列传第一百一十四·王播等》㉕*）&lt;p&gt; 是时，南方旱歉，人相食，（王）播掊敛不少衰，民皆怨之。（《新唐书·卷一百六十七·列传第九十二·白裴崔韦二李皇甫王》㉕）&lt;/p&gt;
#   829年: 属岁旱俭，人至相食，楚均富赡贫，而无流亡者。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）
#   832年:（李）听先遣亲吏至徐州慰劳将士，苍头不欲听复来，说军士杀其亲吏，脔食之。（《资治通鉴》卷244）
#   约841年: （杜牧）作《罪言》。其辞曰：……. 山东叛且三五世，后生所见言语举止，无非叛也，以为事理正当如此，沉酣入骨髓，无以为非者，至有围急食尽，啖尸以战。以此为俗，岂可与决一胜一负哉？（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕*）
#   868年: 其年冬，庞勋杀崔彦曾，据徐州，聚众六七万。徐无兵食，乃分遣贼帅攻剽淮南诸郡，滁、和、楚、寿继陷。谷食既尽，淮南之民多为贼所啖。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）&lt;p&gt; 勋还，果盗徐州，其众六七万。徐乏食，分兵攻滁、和、楚、寿，陷之，粮尽，啖人以饱。（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕）&lt;/p&gt;
#   868年: 一日，贼军乘间，步骑径入湘垒，淮卒五千人皆被生絷送徐州，为贼蒸而食之。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）&lt;/p&gt;&lt;p&gt;湘乃彻警释械，日与勋众欢言。后贼乘间直袭湘垒，悉俘而食之，醢湘及监军郗厚本。（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕）&lt;/p&gt;
#   868年: 庞勋又令将刘贽攻濠州，陷之，囚刺史卢望回于回车馆，望回郁愤而死，仆妾数人皆为贼蒸而食之。（《旧唐书·卷十九上·本纪第十九上·懿宗》㉕*）
#   869年: 吴迥守濠州，粮尽食人，驱女孺运薪塞隍，并填之，整旅而行，马士举斩以献。（《新唐书·卷一百四十八·列传第七十三·令狐张康李刘田王牛史》㉕*）&lt;p&gt;马举攻濠州，自夏及冬不克，城中粮尽，杀人而食之（《资治通鉴》卷251）&lt;/p&gt;
#   876年：李廷节妻崔。乾符中，廷节为郏城尉。王仙芝攻汝州，廷节被执。贼见崔妹美，将妻之，诟曰：“我，士人妻，死亡有命，奈何受贼污？”贼怒，刳其心食之。（《新唐书·卷二百五·列传第一百三十·列女》㉕*）
#   878年: （李）尽忠械文楚等五人送斗鸡台下，（李）克用令军士玼食之，以骑践其骸。（《资治通鉴》卷253）
#   881年，[[:w:唐僖宗|唐僖宗]]廣明二年：（[[:w:黃巢|黃巢]]攻佔長安，）時京畿百姓皆寨于山谷，累年費耕耘，賊坐空城，賦輸無如，谷食騰踴，米斗三十錢，官軍皆執山寨百姓，蠰于賊為食，人獲數十萬”（《[[:w:舊唐書|舊唐書]]·卷二百下·列传第一百五十·朱泚 黄巢 秦宗权》㉕*）&lt;p&gt; 二年春正月甲辰朔，天下勤王之师，云会京畿，京师食尽。贼食树皮，以金玉买人于行营之师，人获数百万。山谷避乱百姓，多为诸军之所执卖。（《旧唐书·卷十九下·本纪第十九下 僖宗》㉕）&lt;/p&gt;&lt;p&gt;于时畿民栅山谷自保，不得耕，米斗钱三十千，屑树皮以食，有执栅民鬻贼以为粮，人获数十万钱。（《新唐书·卷二百二十五下·列传第一百五十下·逆臣下》㉕）&lt;/p&gt;&lt;p&gt;民避乱皆入深山筑栅自保，农事俱废，长安城中斗米直三十缗。贼（黄巢）卖人于官军以为粮，官军或执山栅之民鬻之，人直数百缗，以肥瘠论价。（《资治通鉴》卷254）&lt;/p&gt;
#  883年，唐僖宗中和三年883年：时黄巢与宗权合从，纵兵四掠，远近皆罹其酷。时仍岁大饥，民无积聚，贼俘人为食，其炮炙处谓之“舂磨寨”，白骨山积，丧乱之极，无甚于斯。（《旧唐书·卷十九下·本纪第十九下 僖宗》㉕*）&lt;p&gt;贼（黄巢）围陈郡百日，关东仍岁无耕稼，人饿倚墙壁间，贼俘人而食，日杀数千。贼有舂磨砦，为巨碓数百，生纳人于臼碎之，合骨而食，其流毒若是。（《旧唐书·卷二百下·列传第一百五十 朱泚 黄巢 秦宗权》㉕）&lt;/p&gt;&lt;p&gt;巢已东，使孟楷攻蔡州。节度使秦宗权迎战，大败，即臣贼，与连和。楷击陈州，败死，巢自围之，略邓、许、孟、洛，东入徐、兖数十州。人大饥，倚死墙堑，贼俘以食，日数千人，乃办列百巨碓，糜骨皮于臼，并啖之。（《新唐书·卷二百二十五下·列传第一百五十下 逆臣下》㉕）&lt;/p&gt;&lt;p&gt;是时，陈州四面，贼寨相望，驱掳编氓，杀以充食，号为“舂磨寨”。（《旧五代史·卷一（梁书）·太祖纪一》㉕）&lt;/p&gt;&lt;p&gt;秦宗权以蔡州附巢，巢势甚盛，乃悉众围犨，置舂磨，糜人之肉以为食。（《新五代史·卷四十二·杂传第三十·朱宣等》㉕）&lt;/p&gt;&lt;p&gt;时民间无积聚，贼（黄巢）掠人为粮，生投于碓硙，并骨食之，号给粮之处曰“舂磨寨”。纵兵四掠，自河南、许、汝、唐、邓、孟、郑、汴、曹、濮、徐、兖等数十州，咸被其毒。 （《资治通鉴》卷255）&lt;/p&gt;
#   884年: （秦宗权）所至屠翦焚荡，殆无孑遗。其残暴又甚于巢，军行未始转粮，车载盐尸以从。北至卫、滑，西及关辅，东尽青、齐，南出江、淮，州镇存者仅保一城，极目千里，无复烟火。（《资治通鉴》卷256）&lt;p&gt; 巢贼虽平，而宗权之凶徒大集，西至金、商、陕、虢，南极荆、襄，东过淮甸，北侵徐、兖、汴、郑，幅员数十州。五六年间，民无耕织，千室之邑，不存一二，岁既凶荒，皆脍人而食，丧乱之酷，未之前闻。（《旧唐书·卷二十上·本纪第二十上·昭宗》㉕*）&lt;/p&gt;&lt;p&gt;（秦宗权）贼首皆慓锐惨毒，所至屠残人物，燔烧郡邑。西至关内，东极青、齐，南出江淮，北至卫滑，鱼烂鸟散，人烟断绝，荆榛蔽野。贼既乏食，啖人为储，军士四出，则盐尸而从。（《旧唐书·卷二百下·列传第一百五十·朱泚 黄巢 秦宗权》㉕）&lt;/p&gt;&lt;p&gt; 中和二年，关内大饥。四年，关内大饥，人相食。（《新唐书·卷三十五·志第二十五 稼穑不成》㉕）&lt;/p&gt;&lt;p&gt;中和四年，江南大旱，饥，人相食。（《新唐书·卷三十五·志第二十五·常旸》㉕）&lt;/p&gt;
#   886年: 荆南、襄阳仍岁蝗旱，米斗三十千，人多相食。（《旧唐书·卷十九下·本纪第十九下·僖宗》㉕*）&lt;p&gt; 光启二年二月，荆、襄大饥，米斗三千钱，人相食。（《新唐书·卷三十五·志第二十五·稼穑不成》㉕）&lt;/p&gt;&lt;p&gt;二年，荆、襄蝗、米斗钱三千，人相食；（《新唐书·卷三十六·志第二十六·五行三》㉕）&lt;/p&gt;
#   886年: （张）瑰固垒二岁，樵苏皆尽，米斗钱四十千，计抔而食，号为“通肠”。疫死者，争啖其尸，县首于户以备馔。（《新唐书·卷一百八十六·列传第一百一十一 ·周王邓陈齐赵二杨顾》㉕*）
#   887年: 戊午，秦彦遣毕师铎、秦稠将兵八千出（扬州）城，西击杨行密。稠败死，士卒死者什七八。城中乏食，樵采路绝，宣州军始食之。（《资治通鉴》卷257）&lt;p&gt;五月，寿州刺史杨行密率兵攻（秦）彦，……重围半年，（扬州）城中刍粮并尽，草根木实、市肆药物、皮囊革带，食之亦尽。外军掠人而卖，人五十千。死者十六七，纵存者鬼形鸟面，气息奄然。（《旧唐书·卷一百八十二·列传第一百三十二·王重荣等》㉕*）&lt;/p&gt;&lt;p&gt;杨行密围扬州，毕师铎厚赍宝币，啖（杜）雄连和。雄率军浮海屯东塘。是时扬州围久，皮囊革带食无余，军中杀人代粮，才千钱。（《新唐书·卷一百九十·列传第一百一十五·三刘成杜钟张王》㉕）&lt;/p&gt;&lt;p&gt;是时，城中仓廪空虚，饥民相杀而食，其夫妇、父子自相牵，就屠卖之，屠者刲剔如羊豕。（《新五代史·卷六十一·吴世家第一》㉕）&lt;/p&gt;
#   887年: （高）骈家属并在道院，秦彦供给甚薄，薪蒸亦阙。奴仆彻延和阁栏槛煮革带食之，互相篡啖。（《旧唐书·卷一百八十二·列传第一百三十二·王重荣等》㉕*）&lt;p&gt;高骈在道院，秦彦供给甚薄，左右无食，至然木像、煮革带食之，有相啖者。（《资治通鉴》卷257）&lt;/p&gt;
#   887年，光启三年：（杨）行密攻围（广陵）弥急，城中食尽，米斗四十千，居人相啖略尽。十月，城陷，秦、毕走东塘，行密入广陵，辇外寨之粟以食饥民，即日米价减至三千。（《旧五代史·卷一百三十四·僭伪列传一》㉕*）&lt;p&gt;[[:w:杨行密|杨行密]]围广陵且半年，秦彦、毕师铎大小数十战多不利，城中无食，料值钱五十缗，草根木实皆尽，以堇泥为饼食之，饿死者大半。宣州军掠人诣肆卖之，驱缚屠割如羊豕，讫无一声，流血满于坊市。彦、师铎无如之何，颦蹙而已。（《资治通鉴》卷257）&lt;/p&gt;
#   887年: 周迪妻某氏。迪善贾，往来广陵。会毕师铎乱，人相掠卖以食。迪饥将绝，妻曰：“今欲归，不两全。君亲在，不可并死，愿见卖以济君行。”迪不忍，妻固与诣肆，售得数千钱以奉。迪至城门，守者谁何，疑其绐，与迪至肆问状，见妻首已在枅矣。迪里余体归葬之。（《新唐书·卷二百五·列传第一百三十·列女》㉕*）
#   888年: （李）罕之与（张）言甚笃，然性猜暴。是时大乱后，野无遗秆，部卒日剽人以食。《新唐书·卷一百八十七·列传第一百一十二·二王诸葛李孟》㉕*）&lt;p&gt;时大乱之后，野无耕稼，罕之部下以俘剽为资，啖人作食。……自是罕之日以兵寇钞怀、孟、晋、绛，数百里内，郡邑无长吏，闾里无居民。……自是数州之民，屠啖殆尽，荆棘蔽野，烟火断绝，凡十余年。（《旧五代史·卷十五（梁书）·列传五》㉕）&lt;/p&gt;&lt;p&gt;罕之留其子颀事晋，乃之泽州，日以兵钞怀、孟间，啖人为食。（《新五代史·卷四十二·杂传第三十·朱宣等》㉕）&lt;/p&gt;&lt;p&gt;（李）罕之勇而无谋，性复贪暴，意轻（张）全义，闻其勤俭力穑，笑曰：“此田舍一夫耳！”…….（李）罕之所部不耕稼，专以剽掠为资，啖人为粮。……（李罕之）以寇钞为事，自怀、孟、晋、绛数百里间，州无刺史，县无令长，田无麦禾，邑无烟火者，殆将十年。（《资治通鉴》）&lt;/p&gt;
#   889年，[[:w:唐昭宗|唐昭宗]]龍紀元年：楊行密圍宣州，城中食盡，人相啖……（《資治通鑒》卷258）
#   891年: 会吏盗减诸军禀食，（王）建怒其众曰：“招讨吏之谋也。”纵士执之，醢食于军。（《新唐书·卷二百二十四下·列传第一百四十九下·叛臣下》㉕*）&lt;p&gt;一日，（王）建阴令军士于行府门外擒（韦）昭度亲吏，脔而食之，（王）建徐启（韦）昭度曰：“盖军士乏食，以至于是耶！”昭度大惧，遂留符节与建，即日东还。（《旧五代史·卷一百三十六·僭伪列传三》㉕）&lt;/p&gt;&lt;p&gt;昭度迟疑未决，建遣军士擒昭度亲吏于军门，脔而食之，建入白曰：“军士饥，须此为食尔！”昭度大恐，即留符节与建而东。（《新五代史·卷六十三·前蜀世家第三》㉕）&lt;/p&gt;&lt;p&gt;庚子，（王）建阴令东川将唐友通等擒（韦）昭度亲吏骆保于行府门，脔食之，云其盗军粮。（《资治通鉴》卷258）&lt;/p&gt;
#   891年: 孙儒悉焚扬州庐舍，尽驱丁壮及妇女渡江，杀老弱以充食。（《资治通鉴》卷258）
#   893年: 景福二年春，（李克用）大举以伐王镕，……王镕出师三万来援，武皇（李克用）逆战于叱日岭下，镇人败，斩首万余级。时岁饥，军乏食，脯尸肉而食之。（《旧五代史·卷二十六（唐书）·武皇纪下》㉕*）&lt;p&gt;（李克用的）河东军无食。脯其尸而啖之。 （《资治通鉴》卷259）&lt;/p&gt;
#   894年: 王建攻彭州，城中人相食（《资治通鉴》卷259）
#   902年，唐昭宗天复二年：是冬，大雪，（凤翔）城中食尽，冻馁死者不可胜计，或卧未死，肉已为人所。市中卖人肉斤直钱百，犬肉值五百。”（《资治通鉴》卷263）&lt;p&gt;昭宗在凤翔，为梁兵所围，城中人相食，父食其子，而天子食粥，六宫及宗室多饿死。其穷至于如此，遂以亡。（《新唐书·卷五十二·志第四十二·食货二》㉕*）&lt;/p&gt;&lt;p&gt;（朱温的后）梁军围之（凤翔）逾年，（李）茂贞每战辄败，闭壁不敢出。城中薪食俱尽，自冬涉春，雨雪不止，民冻饿死者日以千数。米斗直钱七千，至烧人屎煮尸而食。父自食其子，人有争其肉者，曰：“此吾子也，汝安得而食之！”人肉斤直钱百，狗肉斤直钱五百。父甘食其子，而人肉贱于狗。天子于宫中设小磨，遣宫人自屑豆麦以供御，自后宫、诸王十六宅，冻馁而死者日三四。城中人相与邀遮茂贞，求路以为生。（《新五代史·卷四十·杂传第二十八·李茂贞等》㉕）&lt;/p&gt;

==五代十國==
#   906年:天祐三年，（朱）全忠自将攻沧州，……全忠环沧筑而沟之，内外援绝，人相食。（刘）仁恭求战，不许。（《新唐书· 卷二百一十二·列传第一百三十七·藩镇卢龙》㉕*）&lt;p&gt;汴人深沟高垒以攻沧州，内外阻绝，（刘）仁恭不能合战，城中大饥，人相篡啖，析骸而爨，丸土而食，转死骨立者十之六七。……城中乏食，米斗直三万，人首一级亦直十千，军士食人，百姓食墐土，驴马相遇，食其鬃尾，士人出入，多为强者屠杀。（《旧五代史·卷一百三十五·僭伪列传二》㉕）&lt;/p&gt;&lt;p&gt;梁军壁长芦，深沟高垒，（刘）仁恭不能近。沧州被围百余日，城中食尽，人自相食，析骸而爨，或丸墐土而食，死者十六七。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕）&lt;/p&gt;&lt;p&gt;时汴军筑垒围沧州，鸟鼠不能通。（刘）仁恭畏其强，不敢战。城中食尽，丸土而食，或互相掠啖。（《资治通鉴》卷265）&lt;/p&gt;
#   909年:（刘）守文将吏孙鹤、吕兖等，立守文子延祚以距（刘）守光，守光围之百余日，城中食尽，米斛直钱三万，人相杀而食，或食墐土，马相食其骏尾，（吕）兖等率城中饥民食以麹，号“宰务”，日杀以饷军。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕*）&lt;p&gt;刘守光围沧州久不下，执刘守文至城下示之，犹固守。城中食尽，民食堇泥，军士食人，驴马相啖尾。吕兖选男女羸弱者，饲以黮面而烹之，以给军食，谓之宰杀务。 （《资治通鉴》卷267）&lt;/p&gt;
#  911: （刘）守光大怒，推之（孙鹤）伏锧，令军士割其肉生啖之。鹤大呼曰：“百日之外，必有急兵矣！”守光命窒其口，寸斩之，有识为之嗟惋。（《旧五代史·卷一百三十五·僭伪列传二》㉕*）&lt;p&gt;（刘）守光怒，推之（孙鹤）伏锧，令军士割而啖之。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕）&lt;/p&gt;&lt;p&gt;（刘）守光怒，伏诸质上，令军士剐而啖之。鹤呼曰：“百日之外，必有急兵！”守光命以土窒其口，寸斩之。（《资治通鉴》卷268）&lt;/p&gt;
#  916: 晋人围贝州逾年，城中食尽，啖人为粮。（《资治通鉴》卷269）
#   922年: （李存勖）获（张）处球、处瑾、处琪并其母，及同恶高濛李翥、齐俭等，皆折足送行台，镇人请醢而食之；（《旧五代史·卷二十九（唐书）·庄宗纪三》㉕*）
#  925年，後唐莊宗同光三年： （郭）崇韬欲诛（王）宗弼以自明，己巳，白（李）继岌收宗弼及王宗勋、王宗渥，皆数其不忠之罪，族诛之，籍没其家。蜀人争食宗弼之肉。 （《资治通鉴》卷274）
#   929年: （董璋）遣其将李彦钊扼剑门关为七砦，于关北增置关，号永定。凡唐戍兵东归者，皆遮留之，获其逃者，覆以铁笼，火炙之，或刲肉钉面，割心而啖。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕*）
#  930: （董）璋怒，令军士十人，持刀刲割其（姚洪）肤，燃镬于前，自取啖食，洪至死大骂不已。（《旧五代史·卷七十（唐书）·列传二十二》㉕*）&lt;p&gt;（董）璋怒，然镬于前，令壮士十人刲其肉而食，洪至死大骂。（《新五代史·卷三十三·死事传第二十》㉕）&lt;/p&gt;&lt;p&gt;（董）璋怒，然镬于前，令壮士十人刲其（姚洪）肉自啖之，洪至死骂不绝声。（《资治通鉴》卷277）&lt;/p&gt;
#   约930年:（李）赞华好饮人血，姬妾多刺臂以吮之；婢仆小过，或抉目，或刀刲火灼；夏氏不忍其残，奏离婚为尼。 （《资治通鉴》卷277）
#  934: （薛）文杰善数术，自占云：“过三日可无患。”送者闻之，疾驰二日而至，军士踊跃，磔文杰于市，闽人争以瓦石投之，脔食立尽。（《新五代史·卷六十八·闽世家第八》㉕*）&lt;p&gt;（薛）文杰出，（王）继鹏伺之于启圣门外，以笏击之仆地，槛车送军前，市人争持瓦砾击之。文杰善术数，自云过三日则无患。部送者闻之，倍道兼行，二日而至，士卒见之踊跃，脔食之（《资治通鉴》卷278）&lt;/p&gt;
#   约942年: （石）信所至黩货，好行杀戮。军士有犯法者，信召其妻子，对之刲剔支解，使自食其肉，血流盈前，信命乐饮酒自如也。（《新五代史·卷十八·汉家人传第六》㉕*）
#   944年: 同（州）、华（州）奏，人民相食。（《旧五代史·卷七十（唐书）·列传二十二》㉕*）
#   944年: （后晋少帝石重贵）命李守贞、符彦卿率师东讨。（杨）光远素无兵众，惟婴城（青州）自守，守贞以长连城围之。冬十一月，（杨）承勋与弟承信、承祚见城中人民相食将尽，知事不济，劝（杨）光远乞降，冀免于赤族。（《旧五代史·卷九十七（晋书）·列传十二》㉕*）&lt;p&gt;契丹已北，出帝（石重贵）复遣（李守贞、符彦卿东讨，光远婴城固守，自夏至冬，城中人相食几尽。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕）&lt;/p&gt;
#  945年: 闽人或告福州援兵谋叛，闽主（王）延政收其铠仗，遣还，伏兵于隘，尽杀之，死者八千馀人，脯其肉以归为食。 （《资治通鉴》卷284）
#  947年: （杨）承勋事晋为郑州防御使，（耶律）德光灭晋，使人召承勋至京师，责其劫父，脔而食之。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕*）&lt;p&gt;戊子，（辽军）执郑州防御使杨承勋至大梁，责以杀父叛契丹，命左右脔食之。（《资治通鉴》卷286）&lt;/p&gt;
#  947年，后晋天福十二年（947年：大同元年春正月……己丑，以张彦泽擅徙重贵开封，杀桑维翰，纵兵大掠，不道，斩于市。晋人脔食之。（《辽史· 卷四·本纪第四·太宗下》㉕*）&lt;p&gt;戎王（辽太宗耶律德光）知其（张彦泽）众怒，遂令弃市，仍令高勋监决，断腕出锁，然后刑之。勋使人剖其心以祭死者，市人争其肉而食之。（《旧五代史·卷九十八（晋书）·列传十三》㉕）&lt;/p&gt;&lt;p&gt;百官皆请不赦（张彦泽），而都人争投状疏其恶，乃命高勋监杀之。彦泽前所杀士大夫子孙，皆缞绖杖哭，随而诟詈，以杖朴之，彦泽俯首无一言。行至北市，断腕出锁，然后用刑，勋剖其心祭死者，市人争破其脑，取其髓，脔其肉而食之。（《新五代史·卷五十二·杂传第四十·杜重威等》㉕）&lt;/p&gt;&lt;p&gt;己丑，斩（张）彦泽、（傅）住皃于北市，仍命高勋监刑。彦泽前所杀士大夫子孙，皆绖杖号哭，随而诟詈，以杖扑之。勋命断腕出锁，剖其心以祭死者。市人争破其脑取髓，脔其肉而食之。 （《资治通鉴》卷286）&lt;/p&gt;
#  948年: （苏）逢吉等秘不发丧，下诏称：“（杜）重威父子，因朕小疾，谤议摇众，皆斩之。”磔死于市，市人争啖其肉。（《旧五代史·卷一百（汉书）·高祖纪下》㉕*）&lt;p&gt;磔（杜）重威尸于市，市人争啖其肉，吏不能禁，斯须而尽。 （《资治通鉴》卷287）&lt;/p&gt;
#   948年: （李）守贞自谓天时人事合符于己，乃潜结草贼，令所在窃发，遣兵据潼关。朝廷命白文珂、常思等领兵问罪，复遣枢密使郭威西征。……既而城中粮尽，杀人为食。（《旧五代史·卷一百九（汉书）·列传六》㉕*）&lt;p&gt;（李）守贞（潼关）城中兵无几，而食又尽，杀人而食。（《新五代史·卷五十二·杂传第四十·杜重威等》㉕）&lt;/p&gt;
#  949年，後漢高祖乾佑元年二年：（赵）思绾粮尽，城中人相食（宋）（《宋史· 卷二百五十二·列传第十一·王景等》㉕*）&lt;p&gt;朝廷闻之，命郭从义、王峻帅师伐之（赵思绾）。及攻其城（长安），王师伤者甚众，乃以长堑围之。经年粮尽，遂杀人充食。思绾尝对众取人胆以酒吞之，告众曰：“吞此至一千，即胆气无敌矣。”（《太平广记》：贼臣赵思绾自倡乱至败，凡食人肝六十六，无不面剖而脍之。）（《旧五代史·卷一百九（汉书）·列传六》㉕）&lt;/p&gt;&lt;p&gt;隐帝（后汉隐帝刘承祐）遣郭威西督诸将兵，先围（李）守贞于河中。居数月，（赵）思绾城中食尽，杀人而食，每犒宴，杀人数百，庖宰一如羊豕。思绾取其胆以酒吞之，语其下曰：“食胆至千，则勇无敌矣！” （《新五代史·卷五十三·杂传第四十一·王景崇等》㉕）&lt;/p&gt;&lt;p&gt;赵思绾好食人肝，常面剖而脍之，脍尽，人犹未死。又好以酒吞人胆，谓人曰：吞此千数，则胆无敌矣。长安城中食尽，取妇女幼稚为军粮，日计数而给之。每犒军，辄屠数百人，如羊豖法。（《资治通鉴》卷288）&lt;/p&gt;
#  950年: （马希萼）脔食李弘皋、（李）弘节、唐昭胤、杨涤。（《资治通鉴》）
#   苌从简（后唐、后晋武将），陈州人也。……好食人肉，所至多潜捕民间小儿以食。（《新五代史·卷四十七·杂传第三十五·华温琪等》㉕*）
#  [[:w:吴国 (五代十国)|吳國]]將領[[:w:高澧|高澧]]「嗜殺人而飲血，日暮，必於宅前，後掠行人而食之」。（《南村辍耕录》引《九国志》）

==辽宋金==
从《宋史》开始，二十五史开始频繁记载割肉疗亲的尽孝的故事，这反映了儒家伦理和人肉治病理念的普及，宋朝官方是褒奖这种做法的，之后元朝法律禁止，明清官方态度有所保留，但屡禁不止，愈演愈烈。

* 冠冕百行莫大于孝，范防百为莫大于义。先王兴孝以教民厚，民用不薄；兴义以教民睦，民用不争。率天下而由孝义，非履信思顺之世乎。太祖、太宗以来，子有复父仇而杀人者，壮而释之；刲股割肝，咸见褒赏；至于数世同居，辄复其家。一百余年，孝义所感，醴泉、甘露、芝草、异木之瑞，史不绝书，宋之教化有足观者矣。作《孝义传》。《宋史· 卷四百五十六·列传第二百一十五·孝义》

岳飞《满江红》的“壮志饥餐胡虏肉，笑谈渴饮匈奴血”可能是大众文化中最广泛流传的称赞吃人的文学作品。

#  辽穆宗时期（951年-969年）：初，女巫肖古上延年药方，当用男子胆和之。不数年，杀人甚多，至是（957年，应历七年），觉其妄，辛巳，射杀之。（《辽史·卷六·本纪第六·穆宗上》㉕*）&lt;p&gt;京师置百尺牢以处系囚。盖其（辽穆宗）即位未久，惑女巫肖古之言，取人胆合延年药，故杀人颇众。后悟其诈，以鸣镝丛射、骑践杀之。（《辽史·卷六十一·志第三十·刑法志上》㉕）&lt;/p&gt;
#   963年: 众皆感愤，遂破其众于平津亭，擒（张）文表脔而食之。（《宋史· 卷四百八十三·列传第二百四十二·世家六》㉕*）
#  963年乾德元年：（李）处耘释所俘体肥者数十人，令左右分啖之，黥其少健者，令先入朗州。 （《宋史· 卷二百五十七·列传第十六· 吴廷祚等》㉕*）
#  969年，開寶二年（969）：[[:w:王彥昇|王彥昇]]改防州防御使，是冬，又移原州（甘肅鎮原）。 西人（甘肅少數民族）有犯漢法者，彥升不加刑，召僚屬飲宴，引所犯，以手捽斷其耳，大嚼，巵酒下之。其人流血被體，股栗不敢動。前後啗者數百人。西人畏之，不敢犯塞。（[[:w:王辟之|王辟之]]《澠水燕談錄》，《宋史·卷二百五十·列传第九·王彥昇》㉕*）
#  970年，开宝三年：命分司西京。（王）继勋残暴愈甚，强市民家子女备给使，小不如意，即杀食之，而棺其骨弃野外。……长寿寺僧广惠常与继勋同食人肉，令折其胫而斩之。洛民称快。（《宋史· 卷四百六十三·列传第二百二十二·外戚上》㉕*）
#  1006年: 三年，（德恭）被疾，子承庆刲股肉食之。（《宋史· 卷二百四十四·列传第三·宗室一》㉕*）
#  1048年，[[:w:宋仁宗|宋仁宗]]庆历八年：明年，河北大饥，人相食，（子）鼎经营赈救，颇尽力。（《宋史·卷三百·列传第五十九·杨偕等》㉕*）&lt;p&gt;河北、京東西大水為災，人相食，流民入京東者不可勝數（《[[:w:續資治通鑑|續資治通鑑]]》卷50）&lt;/p&gt;
#  约1053年，宋仁宗时期：[[:w:侬智高|（侬）智高]]母[[:w:阿侬|阿侬]]有计谋，智高攻陷城邑，多用其策，僭号皇太后，性惨毒，嗜小儿肉，每食必杀小儿。（《宋史· 卷四百九十五·列传第二百五十四·蛮夷三》㉕*） 
#  1087年，[[:w:宋哲宗|宋哲宗]]元祐二年，[[:w:苏辙|苏辙]]《因旱乞许群臣面对言事剳子》：“臣伏见二年以来，民气未和，天意未顺，災沴荐至，非水即旱。淮南饥饉，人至相食。河北流移，道路不绝。京东困弊，盗贼群起。二圣遇災忧惧，顷发仓廪以救其乏绝，独此三路所散，已仅三百万斛矣！異时赈賉未见此比。然而民力已困，国用己竭，而旱势未止，夏麦失望，秋稼未立，数月之后，公私无继，群盗蜂起，势有必至，臣未知朝廷何以待此？……”
#  1102年: （高永年）行三十里，逢羌帐下亲兵，皆永年昔所推纳熟户也。永年不之备，羌遽执永年以叛，遂为多罗巴所杀，探其心肝食之，谓其下曰：“此人夺我国，使吾宗族漂落无处所，不可不杀也。”（《宋史· 卷四百五十三·列传第二百一十二·忠义八》㉕*）
#  1118年，辽天庆八年（宋重和元年，1118年），十二月，“宁昌军（治懿州）节度使刘宏（无可考）以懿州（治宁昌，今阜新市东北之塔营子村）户三千降金。时山前诸路（此指辽东，非燕山之南）大饥，乾（辽宁北镇南）显（北镇北）宜（义县）锦（锦州市）兴中（朝阳市）等路，斗粟值数缣，民削榆皮食之，既而人相食。”（《辽史· 卷二十八·本纪第二十八·天祚皇帝二》㉕*）
#  1121年: 贼（霍成富）怒，脔其（詹良臣）肉，使自啖之。良臣吐且骂，至死不绝声，见者掩面流涕，时年七十二。（《宋史· 卷四百四十六·列传第二百五·忠义一》㉕*）
#  “甲辰宣和六年（1124年）时转粮给燕山（府治北京西南）民力疲困，重以盐额科敛，加之连年凶荒，民食榆皮野菜不给，至自相食。于是饥民并起为盗。山东有张万仙者，众十万，号敢炽。张迪者，众五万，围濬州（濬州，平川军，治滑州黎阳）五日而去。濬州去京纔一百六十里，而初不知。河北有高托山者，号三十万。其余一二万者，不可胜计也。”（《九朝编年备要卷二十九》）
#  [[:w:宋徽宗|宋徽宗]]宣和七年（1125年）十二月，金两路攻宋。王禀皆破之，“然人众乏粮，三军先食牛马骡，次烹弓弩皮甲，百姓煮萍实、糠籺、草茭以充腹，既而人相食。[九月]城破，禀犹率羸卒巷战，突围出，金兵追之急，遂负太原庙中太宗御容赴汾水死，子荀殉之。”（《续资治通鉴卷九十七》）
#  1125年: 刘敏行，平州人。登天会三年进士。除太子校书郎，累迁肥乡令。岁大饥，盗贼掠人为食。诸县老弱入保郡城，不敢耕种，农事废，畎亩荒芜。（《金史· 卷一百二十八·列传第六十六·循吏》㉕*）
#  1129年：（建炎）三年，山东郡国大饥，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1131年: 有孙知微者，以朝请大夫通判舒州。绍兴元年，贼刘忠入其境，执知微以去，知微不屈，忠怒，脔而食之。（《宋史· 卷四百五十三·列传第二百一十二·忠义八》㉕*）
#  1131年：五湖捕鱼人夏宁聚众千余，掠人为食，郭仲威余党出没淮南，邵青据通州，光世皆招降之。（《宋史·卷三百六十九·列传第一百二十八·张俊》㉕*）&lt;p&gt;五湖捕魚人夏寧，“聚其徒為盜，後有眾千餘，專掠人以為食，……寧等無食，半月之間復啖萬餘人，是日，始具舟迎之。由是江北鄉村愈覺凋殘矣。”（《续资治通鉴卷一零九》）&lt;/p&gt;
#  约1133年，宋高宗紹興三年：唐初，贼朱粲以人为粮，置捣磨寨，谓“啖醉人如食糟豚”。每览前史，为之伤叹。而自靖康丙午岁，金人乱华，六七年间，山东、京西、淮南等路，荆榛千里，斗米至数十千，且不可得。盗贼、官兵以至居民，更互相食。人肉之价，贱于犬豕，肥壮者一枚不过十五千，全躯暴以为腊。登州范温率忠义之人，绍兴癸丑岁泛海到钱唐，有持至行在犹食者。老瘦男子 词谓之“饶把火”，妇人少艾者名为“不羡羊”，小儿呼为“和骨烂”，又通目为“两脚羊”。唐止朱粲一军，今百倍于前世，杀戮焚溺饥饿疾疫陪堕，其死已众，又加之以相食。杜少陵谓“丧乱死多门”，信矣！不意老眼亲见此时，呜呼痛哉！ （莊綽《雞肋編》）
#  [[:w:宋宁宗|宋宁宗]]嘉定年間，[[:w:林千之|林千之]]任西欽州知州，得了一种病（末疾），有個醫士告訴他，吃童女的肉可以強筋健骨。于是，林千之派人在本州境內捕少女，制成肉乾，叫做“地雞”。&lt;ref&gt;王永寬《中國古代酷刑》&lt;/ref&gt;
#  1210年:（嘉定）三年春，建康府大飢，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1215年: 此數人者（李全等造反者），出沒島崓，寶貨山委而不得食，相率食人。（《宋史· 卷四百七十六·列傳第二百三十五·叛臣中》㉕*）
#  1215年: 乙亥，中都降。（王）檝进言曰：“国家以仁义取天下，不可失信于民，宜禁虏掠，以慰民望。”时城中绝粒，人相食，乃许军士给粮，入城转粜，故士得金帛，而民获粒食。（《元史· 卷一百五十三·列传第四十·刘敏等》㉕*）
#  1216: 是春，河朔人相食。（《金史· 卷二十三·志第四·五行》㉕*）&lt;p&gt;四年，河北行省侯摯言：“河北人相食，觀、滄等州鬥米銀十餘兩。（《金史· 卷五十·志第三十一·食貨五》㉕）&lt;/p&gt;&lt;p&gt;金人迁汴，河朔盗起，……太师、国王木华黎兵至城下，……是时兵乱，民废农耕，所在人相食。（《元史· 卷一百五十一·列传第三十八·薛塔剌海等》㉕）&lt;/p&gt;
#  1216年: 邸顺，保定行唐人，岁甲戌，（邸顺）率众来归（元），（元）太祖授行唐令。……丙子，真定饥，群盗据城叛，民皆穴地以避之，盗发地而啖其人，顺擒数百人杀之。（《元史· 卷一百五十一·列传第三十八·薛塔剌海等》㉕*）
#  1224: 十一月……壬子，京城人相食。癸醜，詔曹門、宋門放士民出就食。（《金史· 卷十八·本紀第十八·哀宗下》㉕*）
#  1227年: 时（李）全在围一年，食牛马及人且尽，将自食其军。初军民数十万，至是余数千矣。（《宋史· 卷四百七十七·列传第二百三十六·叛臣下》㉕*）
#  1228年: （完颜）白撒辈纵军四出，剽掠俘虏，挑掘焚炙，靡所不至。哭声相接，尸骸盈野。都尉高禄谦、苗用秀辈仍掠人食之，而白撒诛斩在口，所过官吏残虐不胜，一饭之费有数十金不能给者，公私皇皇，日皆徯大兵至矣。（《金史· 卷一百十三·列传第五十一·完颜赛不等》㉕*）
#  1232年: 时汴京内外不通，米升银二两。百姓粮尽，殍者相望，缙绅士女多行乞于市，至有自食其妻子者，至于诸皮器物皆煮食之，贵家第宅、市楼肆馆皆撤以爨。（《金史· 卷一百十五·列传第五十三·完颜奴申等》㉕*）
#  1233年，绍定六年（1233年）：（南宋大将[[:w:史嵩之|史嵩之]]围唐州，）城中粮尽，人相食，金将乌库哩黑汉，杀其爱妾以啖士，士争杀其妻子（《金史· 卷一百二十三·列传第六十一·忠义三》㉕*，《续资治通鉴·宋纪》）&lt;p&gt;乙酉，大元召宋兵攻唐州，元帅右监军乌古论黑汉死于战，主帅蒲察某为部曲兵所食。城破，宋人求食人者尽戮之，余无所犯。（《金史· 卷十八·本纪第十八·哀宗下》㉕）&lt;/p&gt;
#  1233: 国用安，先名安用，本名咬儿，淄州人。红袄贼杨安儿、李全余党也。……移兵攻徐，（国）用安投水死，求得其尸，剖面系马尾，为怨家田福一军脔食而尽。（《金史· 卷一百十七·列传第五十五·徒单益都等》㉕*）
#  1234年: 端平元年正月辛丑，黑气压（蔡州）城上，日无光，降者言：“城中绝粮已三月，鞍靴败鼓皆糜煮，且听以老弱互食，诸军日以人畜骨和芹泥食之，又往往斩败军全队，拘其肉以食，故欲降者众。”（《宋史· 卷四百一十二·列传第一百七十一·孟珙》㉕*）
#  1234年:甲午，蔡州破，金主自焚死。时汴梁受兵日久，岁饥，人相食，速不台下令纵其民北渡以就食。（《元史· 卷一百二十一·列传第八·速不台》㉕*）
#  约1237: 岁大饥，人相食，留守别之杰讳不诘，（徐）鹿卿命掩捕食人者，尸诸市。（《宋史· 卷四百二十四·列传第一百八十三·陆持之》㉕*）
#  1272年:咸淳七年，江南大饥。八年冬，襄阳饥，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1276: 德祐二年正月，扬州饥。三月，扬州谷价腾踊，民相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）&lt;p&gt;阿术攻扬（州）久不拔，乃筑长围困之。冬，城中食尽，死者满道。明年二月，饥益甚，赴濠水死者日数百，道有死者，众争割啖之立尽。……兵有烹子而食者，犹日出苦战。（《宋史·卷四百二十一·列传第一百八十·杨栋等》㉕）&lt;/p&gt;
#  1277: 十一月，泸州食尽，人相食，遂破之，安抚王世昌自经死。（《宋史· 卷四百五十一·列传第二百一十·忠义六》㉕*）
#  益州双流人周善敏，丧父，庐于墓侧。母病，又割股肉以啖之，遂愈。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  杨庆，鄞人。父病，贫不能召医，乃刲股肉啖之，良已。其后母病不能食，庆取右乳焚之，以灰和药进焉，入口遂差，久之乳复生。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  （伊）审征幼以孝闻，母病，割股肉啖之。（《宋史· 卷四百七十九·列传第二百三十八·世家二》㉕*）
#  刘孝忠，并州太原人。母病经三年，孝忠割股肉、断左乳以食母；（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  吕升，莱州人。父权失明，剖腹探肝以救父疾，父复能视而升不死。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  成象，渠州流江人。以诗书训授里中，事父母以孝闻。母病，割股肉食之，诏赐束帛醪酒。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  庞天祐，江陵人。以经籍教授里中。父疾，天祐割股肉食之；（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  张伯威，大安军人。……大母黄，年九十八，不忍之官。黄得血痢疾濒殆，伯威剔左臂肉食之，遂愈。继母杨因姑病笃，惊而成疾，伯威复剔臂肉作粥以进，其疾亦愈。伯威妹嫁崔均，其姑王疾，妹亦剔左臂肉作粥以进，达旦即愈。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  母病，（奎）辄割股肉和药以进，母遂愈。（《宋史· 卷三百二十四·列传第八十三·石普》㉕*）
#  （张）掞幼笃孝，蕴病，刲股肉以疗。（《宋史· 卷三百三十三·列传第九十二·杨佐等》㉕*）
#  （常）真妻病，子晏割股肉以养母（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  有朱云孙妻刘氏，姑病，云孙刲股肉作糜以进而愈。姑复病，刘亦刲股以进，又愈。尚书谢谔为赋《孝妇诗》。（《宋史· 卷四百六十·列传第二百一十九·列女》㉕*）
#  聂孝女，字舜英，尚书左右司员外郎天骥之长女也。……崔立劫杀宰相，天骥被创甚，日夜悲泣，恨不即死。舜英谒医救疗百方，至刲其股杂他肉以进，而天骥竟死。时京城围久食尽，……葬其父之明日，绝脰而死。一时士女贤之，有为泣下者。（《金史· 卷一百三十·列传第六十八·列女》㉕*）
#   呼延赞，并州太原人。……其子尝病，赞刲股为羹疗之。（《宋史·卷二百七十九·列传第三十八· 王继忠等》㉕*）
#   蒋偕，字齐贤，华州郑县人。幼贫，有立志。父病，尝刲股以疗，父愈，诘之曰：“此岂孝邪？”曰：“情之所感，实不自知也。”（《宋史·卷三百二十六·列传第八十五·景泰等》㉕*）
#   邑人朱氏女刲股愈母疾，人颂传之，以为治化所致。（《宋史·卷三百四十八·列传第一百七·傅楫等》㉕*）
#   甲幼孤多难，母病，刲股以进。（《宋史·卷三百九十七·列传第一百五十六·徐谊等》㉕*）
#   赵葵，字南仲，京湖制置使方之子。……葵母疾，谒告省侍不得，刲股杂药以寄之。母卒，葵求解官，不许，不得已，卒哭复视事。（《宋史·卷四百一十七·列传第一百七十六·乔行简等》㉕*）
#   陈宗，永嘉人。年十六，母蔡病笃，刲股为饵，病愈。已而复病不救，宗一恸而绝。（《宋史·卷四百五十六·列传第二百一十五·孝义》㉕*）
#   吕仲洙女，名良子，泉州晋江人。父得疾濒殆，女焚香祝天，请以身代，刲股为粥以进。（《宋史·卷四百六十·列传第二百一十九·列女》㉕*）

==元==
元朝法律禁止割肉疗亲，“诸为子行孝，辄以割肝、刲股、埋儿之属为孝者，并禁止之。（《元史· 卷一百五·志第五十三·刑法四》）”但《元史》记载了诸多此般事迹，可见屡禁不止，可能也反映了蒙汉的文化差异。

#   1262年:（中统三年），五月庚申，筑环城（济南）围之；甲戌，围合。（李）鋋自是不得复出，……分军就食民家，发其盖藏以继，不足，则家赋之盐，令以人为食。（《元史·卷二百六·列传第九十三·叛臣》㉕*）
#   1301: 行省右丞刘深远征八百媳妇国，此乃得已而不已之兵也。彼荒裔小邦，远在云南之西南又数千里，……深欺上罔下，帅兵伐之，经过八番，纵横自恣，恃其威力，虐害居民，中途变生，所在皆叛。深既不能制乱，反为乱众所制，军中乏粮，人自相食，（《元史·卷一百六十八·列传第五十五·陈祐（天祥）等》㉕*）
#   1308年:（至大元年六月）河南、山东大饥，有父食其子者，以两道没入赃钞赈之。（《元史· 卷二十二·本纪第二十二·武宗一》㉕*）
#  1319年:延佑六年秋七月丙辰，“来安路总管岑世兴叛，据唐兴州”，杀兼州知州[[:w:黄克仁|黄克仁]]，分食其尸。&lt;ref&gt;《新元史·卷二百四十八·列传第一百四十五》；《招捕总录》&lt;/ref&gt;
#   约1329年: 贼稍引去，（褚不华）乃出，抵杨村桥，贼奄至，杀廉访副使不达失里，啖其尸。（《元史· 卷一百九十四·列传第八十一·忠义二》㉕*）
#   约1329年: （褚）不华以余兵入淮安。……城中饿者仆道上，即取啖之，一切草木、螺蛤、鱼蛙、燕乌，及靴皮、鞍韂、革箱、败弓之筋皆尽，而后父子夫妇老稚更相食，撤屋为薪，人多露处，坊陌生荆棘。力既尽，城陷。（《元史· 卷一百九十四·列传第八十一·忠义二》㉕*）
#   1328年: （天历元年十二月）陕西自泰定二年至是岁不雨，大饥，民相食。（《元史· 卷三十二·本纪第三十二·文宗一》㉕*）&lt;p&gt;天历元年八月，陕西大旱，人相食。（《元史· 卷五十·志第三上·五行一》㉕）&lt;/p&gt;
#   1329年: 天历二年，关中大旱，饥民相食。（《元史· 卷一百七十五·列传第六十二·张珪等》㉕*）&lt;p&gt;文宗天历二年三月，屯田总管兼管河渠司事郭嘉议言：“……近因奉元亢旱，五载失稔，人皆相食，流移疫死者十七八。”（《元史· 卷六十五·志第十七上·河渠二》㉕）&lt;/p&gt;&lt;p&gt;天历二年，（乃蛮台）迁陕西行省平章政事。关中大饥，……京兆民掠人而食之，则命分健卒为队，捕强食人者，其患乃已。（《元史· 卷一百三十九·列传第二十六·乃蛮台等》㉕）&lt;/p&gt;
#   1329:（天历二年夏四月）丙辰，行在所遣只儿哈郎等至京师。河南廉访司言：“河南府路以兵、旱民饥，食人肉事觉者五十一人，饿死者千九百五十人，饥者一万七千四百余人。乞弛山林川泽之禁，听民采食，行入粟补官之令，及括江淮僧道余粮以赈。”（《元史· 卷三十三·本纪第三十三·文宗二》㉕*）
#   1338年: 重改至元四年，…. 贼怒，缚景茂于树，脔其肉，使自啖。景茂益愤骂，贼遂以刀决其口，至耳傍，景茂骂不绝声而死。（《元史· 卷一百九十三·列传第八十·忠义一》㉕*）
#   1342年: 二年春正月…..，是月，大同饥，人相食，运京师粮赈之。（《元史· 卷四十·本纪第四十·顺帝三》㉕*）&lt;p&gt;至正二年，彰德、大同二郡及冀宁平晋、榆次、徐沟县，汾州孝义县，忻州皆大旱，自春至秋不雨，人有相食者。（《元史·卷五十一·志第三下·五行二》㉕）&lt;/p&gt;
#   1343年: （至正）三年，卫辉、冀宁、忻州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1344年: （至正四年）六月，河南巩县大雨，伊、洛水溢，漂民居数百家。济宁路兖州，汴梁鄢陵、通许、陈留、临颍等县大水害稼，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1344年: 八月戊午，祭社稷。丁卯，山东霖雨，民饥相食，赈之。（《元史· 卷四十一·本纪第四十一·顺帝四》㉕*）&lt;p&gt;1344年:（至正四年）八月，益都霖雨，饥民有相食者。（《元史·卷五十一·志第三下·五行二》㉕）&lt;/p&gt;
#   1345年: 五年春，东平路须城、东阿、阳谷三县及徐州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1347: 六月，……彰德路大饥，民相食。（《元史· 卷四十一·本纪第四十一·顺帝四》㉕*）
#   1348: 刘秉直，字清臣，大都武清人。至正八年，来为卫辉路总管，……岁大饥，人相食，死者过半，秉直出俸米，倡富民分粟，馁者食之，病者与药，死者与棺以葬。（《元史· 卷一百九十二·列传第七十九·良吏二》㉕*）
#   1349年: （至正）九年春，胶州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#  [[:w:元惠宗|元惠宗]]至正年间，大饑，“淮右军”軍隊開始吃人，“天下兵甲方殷，而淮右之軍嗜食人，以小兒為上，婦女次之，男子又次之。或使坐兩缸間，外逼以火。或於鐵架上生炙。或縛其手足，先用沸湯澆潑，卻以竹帚刷去苦皮。或盛夾袋中，入巨鍋活煮。或卦作事件而淹之。或男子則止斷其雙腿，婦女則特剜其雙乳。酷毒萬狀，不可具言。總名曰「想肉」，以為食之而使人想之也。”&lt;ref&gt;{{Cite web|title=南村輟耕錄 (四部叢刊本)/卷之九 - 維基文庫，自由的圖書館|url=https://zh.wikisource.org/zh-hant/%E5%8D%97%E6%9D%91%E8%BC%9F%E8%80%95%E9%8C%84_(%E5%9B%9B%E9%83%A8%E5%8F%A2%E5%88%8A%E6%9C%AC)/%E5%8D%B7%E4%B9%8B%E4%B9%9D|website=zh.wikisource.org|access-date=2024-05-28|language=zh-Hant}}&lt;/ref&gt;
#   1352年: （至正）十二年，蕲州、黄州大旱，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1353年: 至正十二年，……明年，春夏大饥，人相食，（余阙）乃捐俸为粥以食之，得活者甚众。（《元史· 卷一百四十三·列传第三十·马祖常等》㉕*）
#   1354年: （至正）十四年，怀庆河内县、孟州，汴梁祥符县，福建泉州，湖南永州、宝庆，广西梧州皆大旱。祥符旱魃再见，泉州种不入土，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1354年: 十四年春，浙东台州，江东饶，闽海福州、邵武、汀州，江西龙兴、建昌、吉安、临江，广西静江等郡皆大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1355: 京师大饥，加以疫疠，民有父子相食者。（《元史· 卷四十三·本纪第四十三·顺帝六》㉕*）
#   1358年: 十八年春，莒州蒙阴县大饥，斗米金一斤。冬，京师大饥，人相食，彰德、山东亦如之。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1358年: （至正）十八年春，蓟州旱。莒州、滨州、般阳淄川县、霍州、鄜州、凤翔岐山县春夏皆大旱。莒州家人自相食，岐山人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1358年: 顺德九县民食蝗，广平人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1359年: （至正）十九年，大都霸州、通州，真定，彰德，怀庆，东昌，卫辉，河间之临邑，东平之须城、东阿、阳谷三县，山东益都、临淄二县，潍州、胶州、博兴州，大同、冀宁二郡，文水、榆次、寿阳、徐沟四县，沂、汾二州，及孝义、平遥、介休三县，晋宁潞州及壶关、潞城、襄垣三县，霍州赵城、灵石二县，隰之永和，沁之武乡，辽之榆社、奉元，及汴梁之祥符、原武、鄢陵、扶沟、杞、尉氏、洧川七县，郑之荥阳、汜水，许之长葛、郾城、襄城、临颍，钧之新郑、密县，皆蝗，食禾稼草木俱尽，所至蔽日，碍人马不能行，填坑堑皆盈。饥民捕蝗以为食，或曝干而积之。又罄，则人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   十九年正月至五月，京师大饥，银一锭得米仅八斗，死者无算。通州民刘五杀其子而食之。保定路莩死盈道，军士掠孱弱以为食。济南及益都之高苑，莒之蒙阴，河南之孟津、新安、黾池等县皆大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1359年: 十八年二月，江西陈友谅遣贼党王奉国等，号二十万，寇信州。明年正月，伯颜不花的斤自衢引兵援焉。……时军民唯食草苗茶纸，既尽，括靴底煮食之，又尽，掘鼠罗雀，及杀老弱以食。五月，大破贼兵。（《元史· 卷一百九十五·列传第八十二·忠义三》㉕*）
#   1360: 至正二十年，（丁好礼）遂拜中书参知政事。时京师大饥，天寿节，庙堂欲用故事大宴会，好礼言：“今民父子有相食者，君臣当修省，以弭大患，燕会宜减常度。”不听，乞谢事，乃以集贤大学士致仕，给全俸家居。（《元史· 卷一百九十六·列传第八十三·忠义四》㉕*）
#   1360年: 李仲义妻刘氏，名翠哥，房山人。至正二十年，县大饥，平章刘哈剌不花兵乏食，执仲义欲烹之。仲义弟马儿走报刘氏，刘氏遽往救之，涕泣伏地，告于兵曰：“所执者是吾夫也，乞矜怜之，贷其生，吾家有酱一瓮、米一斗五升，窖于地中，可掘取之，以代吾夫。”兵不从，刘氏曰：“吾夫瘦小，不可食。吾闻妇人肥黑者味美，吾肥且黑，愿就烹以代夫死。”兵遂释其夫而烹刘氏。闻者莫不哀之。（《元史· 卷二百一·列传第八十八·列女二》㉕*）
#   1362年:（至正）二十二年，河南洛阳、孟津、偃师三县大旱，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   萧道寿，京兆兴平人。……母尝有疾，医累岁不能疗，道寿刲股肉啖之而愈。至元八年，赐羊酒，表其门。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   宁猪狗，山丹州人。母年七十余，患风疾，药饵不效，猪狗割股肉进啖，遂愈。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   潭州万户移剌琼子李家奴，九岁，母病，医言不可治，李家奴割股肉，煮糜以进，病乃痊。抚州路总管管如林、浑州民朱天祥，并以母疾刲割股，旌其家。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   孔全，亳州鹿邑人。父成病，刲股肉啖之，愈。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   赵荣，扶风人。母强氏有疾，荣割股肉啖之者三。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   胡伴侣，钧州密县人。其父实尝患心疾数月，几死，更数医俱莫能疗。伴侣乃斋沐焚香，泣告于天，以所佩小刀于右胁傍刲其皮肤，割脂一片，煎药以进，父疾遂瘳，其伤亦旋愈。朝廷旌表其门。（《元史· 卷一百九十八·列传第八十五·孝友二》㉕*）
#   郎氏，湖州安吉人，宋进士朱甲妻也。……家居，养姑甚谨。姑尝病，郎祷天，刲股肉进啖而愈。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   许氏女，安丰人。父疾，割股啖之乃痊。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   秦氏二女，河南宜阳人，逸其名。父尝有危疾，医云不可攻。姊闭户默祷，凿己脑和药进饮，遂愈。父后复病欲绝，妹刲股肉置粥中，父小啜即苏。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   张义妇，济南邹平人，年十八归里人李伍。……张独家居，养舅姑甚至。父母舅姑病，凡四刲股肉救不懈。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   武用妻苏氏，真定人，徙家京师。用疾，苏氏刲股为粥以进，疾即愈。（《元史· 卷二百一·列传第八十八·列女二》㉕*）

==明==
[[:w:李時珍|李時珍]]完成《本草綱目》，他蒐集藥名是為了「凡經人用者，皆不可遺」，「人部」舉凡毛髮、指甲、牙齒、屎尿、唾液、乳汁、眼淚、汗水、人骨、胞衣（[[:w:紫河車|紫河車]]）、體垢、月水、人勢（陰莖）、人膽、結石……皆可入藥。頭髮可治傷寒、肚疼，男性陰毛治蛇咬，人魄（人吊死級的魂魄）可以安神定魄。

明朝没有像元朝一样法律禁止割肉疗亲，但朱元璋和其礼部尚书公开表示不赞同，但此后仍然多次出现，而且得到政府表彰，还有王族如此做，可见此风难止。

* 至（洪武）二十七年九月，山东守臣言：“日照民江伯儿，母疾，割肋肉以疗，不愈。祷岱岳神，母疾瘳，愿杀子以祀。已果瘳，竟杀其三岁儿。”帝大怒曰：“父子天伦至重。《礼》父服长子三年。今小民无知，灭伦害理，亟宜治罪。”遂逮伯儿，仗之百，遣戍海南。因命议旌表例。礼臣（任亨泰）议曰：“人子事亲，居则致其敬，养则致其乐，有疾则医药吁祷，迫切之情，人子所得为也。至卧冰割股，上古未闻。倘父母止有一子，或割肝而丧生，或卧冰而致死，使父母无依，宗祀永绝，反为不孝之大。皆由愚昧之徒，尚诡异，骇愚俗，希旌表，规避里徭。割股不已，至于割肝，割肝不已，至于杀子。违道伤生，莫此为甚。自今父母有疾，疗治罔功，不得已而卧冰割股，亦听其所为，不在旌表例。”制曰：“可。”（《明史·卷一百三十七·列传第二十五·刘三吾等》）

食人事件的记载：

#  [[:w:韩观|韩观]]杀人甚多，御史欲弹劾他。一日，观召御史饮，以人皮为坐褥，耳目口鼻显然，发散垂褥，首披椅后。肴上，设一人首，观以箸取二目食之，曰：“他禽兽目皆不可食，惟人目甚美。”观前席坐，每拿人至，命斩之，不回首视，已而血流满庭。观曰： “此辈与禽兽不异，斩之如杀虎豹耳。”御史战栗失措曰：“公，神人也。”竟不能劾。&lt;ref&gt;《[[s:湧幢小品/09#韓都督應變|湧幢小品 韓都督應變]]》朱国桢&lt;/ref&gt;
#  1385年，洪武十八年：（韩）林儿本起盗贼，无大志，又听命福通，徒拥虚名。诸将在外者率不遵约束，所过焚劫，至啖老弱为粮，且皆福通故等夷，福通亦不能制。（《明史·卷一百二十二·列传第十·郭子兴 韩林儿》㉕*）
#  约1426年，宣德年间：得（朱）有熺掠食生人肝脑诸不法事，于是并免为庶人。（《明史·卷一百十六·列传第四·诸王》㉕*）
#  1454年，景泰五年：景泰五年，广西古丁等洞贼首蓝伽、韦万山等，纠合蛮类，劫掠南宁、上林、武缘诸处。……贼首韦朝威据古田，县官窜会城，遣典史入县抚谕，烹食之。（《明史·卷三百十七·列传第二百五·广西土司》㉕*）
#  1457年，天顺元年：北畿、山東並飢，發塋墓，斫道樹殆盡。父子或相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  约1465年，成化初：成化初，（彭伦）从赵辅，平大藤峡贼。……（彭）伦大会所部目、把缚俘囚，置高竿，集健卒乱射杀之，复割裂肢体，烹啖诸壮士。（《明史·卷一百六十六·列传第五十四·韩观等》㉕*）
#  1484年，成化二十年：是秋，陝西、山西大旱饑，人相食。停歲辦物料，免稅糧，發帑轉粟，開納米事例振之。（《明史·卷十四·本纪第十四·宪宗二》㉕*）&lt;p&gt;又有虎臣者，麟游人。成化中贡入太学。……省亲归，会陕西大饥，……上言：“臣乡比岁灾伤，人相食，由长吏贪残，赋役失均。请敕有司审民户，编三等以定科徭。”从之。（《明史·卷一百六十四·列传第五十二·邹缉等》㉕）&lt;/p&gt;&lt;p&gt;十六年（何乔新）擢右副都御史，巡抚山西。……进左副都御史。……召拜刑部右侍郎。山西大饥，人相食。命往振，活三十余万人，还流冗十四万户。（《明史·卷一百八十三·列传第七十一·何乔新等》㉕）&lt;/p&gt;&lt;p&gt;汪奎，字文灿，婺源人。……（成化）二十一年，星变，偕同官疏陈十事，言：“……山、陕、河、洛饥民多流郧、襄，至骨肉相啖。请大发帑庾振济，消弭他变。”（《明史·卷一百八十·列传第六十八·张宁等》㉕）&lt;/p&gt;
#  1504年，弘治十七年：十七年，淮、扬、庐、凤洊饥，人相食，且发瘗胔（坟墓尸体）以继之。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1518年，正德十三年：佛郎机，近满剌加。正德中，据满剌加地，逐其王。十三年遣使臣加必丹末等贡方物，请封，始知其名。诏给方物之直，遣还。其人久留不去，剽劫行旅，至掠小儿为食。（《明史·卷三百二十五·列传第二百十三·外国六》㉕*）
#  正德五年（1510年）八月，[[:w:刘瑾|刘瑾]]被磔死，凌迟三日，共剐3300余刀。行刑之日，北京鼎沸，百姓相爭以一钱买刘瑾一塊肉，生吞以泄恨。{{Citation needed}}
#  1519年，正德十四年：是岁，淮、扬饥，人相食。（《明史·卷十六·本纪第十六·武宗》㉕*）&lt;p&gt;十四年三月，有诏南巡，（黄）巩上疏曰：……今江、淮大饥，父子兄弟相食。（《明史·卷一百八十九·列传第七十七·李文祥等》㉕）&lt;/p&gt;&lt;p&gt;（吴）一鹏极陈四方灾异，言：“自去年六月迄今二月，其间天鸣者三，地震者三十八，秋冬雷电雨雹十八，暴风、白气、地裂、山崩、产妖各一，民饥相食二。非常之变，倍于往时。愿陛下率先群工，救疾苦，罢营缮，信大臣，纳忠谏，用回天意。”（《明史·卷一百九十一·列传第七十九·毛澄等》㉕）&lt;/p&gt;
#  1524年，嘉靖三年：三年，湖广、河南、大名、临清饥。南畿诸郡大饥，父子相食，道殣相望，臭弥千里。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;（张）汉卿言：“……今东南洊饥，民至骨肉相食，而搜括之令频行，臣等窃以为不可。”（《明史·卷一百九十二·列传第八十·杨慎》㉕）&lt;/p&gt;&lt;p&gt;世宗即位，（韩邦靖）起山西左参议，分守大同。岁饥，人相食，奏请发帑，不许。（《明史·卷二百一·列传第八十九·陶琰等》㉕）&lt;/p&gt;&lt;p&gt;嘉靖四年二月（余珊）应诏陈十渐，其略曰：……近年以来，黄纸蠲放，白纸催征；额外之敛，下及鸡豚；织造之需，自为商贾。江、淮母子相食，兖、豫盗贼横行，川、陕、湖、贵疲于供饷。（《明史·卷二百八·列传第九十六·张芹等》㉕）&lt;/p&gt;&lt;p&gt;嘉靖初，（湛若水）入朝，……明年进侍读，复疏言：“一二年间，天变地震，山崩川涌，人饥相食，殆无虚月。”（《明史·卷二百八十三·列传第一百七十一·儒林二》㉕）&lt;/p&gt;
#  1529年，嘉靖八年：（杨爵）登嘉靖八年进士，授行人。帝方崇饰礼文，（杨）爵因使王府还，上言：“臣奉使湖广，睹民多菜色，挈筐操刃，割道殍食之。（《明史·卷二百九·列传第九十七·杨最等》㉕*）
#  1549年，嘉靖二十八年：有吴国佐者，洪州司特峒寨苗也，….. 其党石纂太称“太保”，合攻上黄堡，诱败参将黄冲霄，追至永从县，杀守备张世忠，炙而啖之。（《明史·卷二百四十七·列传第一百三十五·刘綎等》㉕*）
#  1552年，嘉靖三十一年：宣、大二镇大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1553年，嘉靖三十二年：京师大饥，人相食，米石二两二钱。（《历代社会风俗事物考》引《金垒子》）
#   1557年，嘉靖三十六年：三十六年，辽东大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1559年，嘉靖三十八年八月：以辽东连年饥馑，至有父食死子者，发银糴粟赈之。（《中外历史年表》）
#   1588，万历十六年：十六年，河南饥，民相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#   1591年，万历十九年：（万历）十九年，（子俊民）还理部事。河南大饥，人相食，请发银米各数十万。（《明史·卷二百十四·列传第一百二·杨博等》㉕*）
#   1593年，万历二十二年：二十二年，河南大饥，人相食，命（钟）化民兼河南道御史往振。荒政具举，民大悦。（《明史·卷二百二十七·列传第一百十五·庞尚鹏等》㉕*）&lt;/p&gt;&lt;p&gt;（陈登云）出按河南。岁大饥，人相食。（《明史·卷二百三十三·列传第一百二十一·姜应麟等》㉕）&lt;/p&gt;
#   1601年，万历二十九年：二十九年，两畿饥。阜平县饥，有食其稚子者。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1611年，万历三十九年：马孟祯，字泰符，桐城人。万历二十六年进士。……三十九年夏，怡神殿灾。孟祯言：“二十年来，郊庙、朝讲、召对、面议俱废，通下情者惟章奏。……畿辅、山东、山西、河南，比岁旱饥。民间卖女鬻儿，食妻啖子，铤而走险，急何能择。”（《明史·卷二百三十·列传第一百十八·蔡时鼎等》㉕*）
#  康熙十二年修《青州府志》第20卷载，万历四十三年（1615年），山东青州府推官[[:w:黄槐开|黄槐开]]在一件申文中说：“自古饥年，止闻道殣相望与易子而食、析骸而爨耳。今屠割活人以供朝夕，父子不问矣，夫妇不问矣，兄弟不问矣。剖腹剜心，支解作脍，且以人心味为美，小儿味尤为美。甚有鬻人肉于市，每斤价钱六文者；有腌人肉于家，以备不时之需者；有割人头用火烧熟而吮其脑者；有饿方倒而众刀攒割立尽者；亦有割肉将尽而眼瞪瞪视人者。间有为人所诃禁，辄应曰：&quot;我不食人，人将食我。&quot;愚民恬不为怪，有司法无所施。枭獍在途，天地昼晦。”
#  1616年，万历四十四年：四十四年，山东饥甚，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;今春以来，天鼓两震于晋地，流星昼陨于清丰，地震二十八，天火九，石首雨菽，河内女妖，辽东兵端吐火，即春秋二百四十年间，未有稠于今日者。且山东大昆，人相食，黄河水稽天。（《明史·卷二百五十七·列传第一百四十五·张鹤鸣等》㉕）&lt;/p&gt;&lt;p&gt;“以山东大饥，致母食死儿，夫食死妻，再振之。”（《中外历史年表》）&lt;/p&gt;
#  萬曆四十五年（1617年）連兩年山東大饑，蔡州有人肉市。“中州兄弟两无子，去山东买妾，遇二女，自称姑嫂，骗兄弟往。兄得小姑。小姑私语之曰：汝弟已为我嫂制成肉羹矣。兄急往视，弟头尚扔炕下。兄急诉之县，抵嫂于罪，兄带小姑去。”（《[[:w:棗林杂俎|棗林杂俎]]》）
#  近日福建抽稅太監高采謬聽方士言：食小兒腦千餘，其陽道可復生如故。乃遍買童稚潛殺之。久而事彰聞，民間無肯鬻者，則令人遍往他所盜至送入，四方失兒者無算，遂至激變掣回。此等俱飛天夜叉化身也。&lt;ref&gt;[[s:萬曆野獲編/卷28#食人]]&lt;/ref&gt;
#  约1621年，天启初：天启初，奢崇明反，率众薄城。（董）尽伦偕知州翁登彦固守。贼遣使说降，尽伦大怒，手刃贼使，抉其睛啖之，屡挫贼锋，城获全。（《明史·卷二百九十·列传第一百七十八·忠义二》㉕*）
#  1622年，天启二年：万化亦率苗仲九股陷龙里，遂围贵阳，自称罗甸王，时天启二年二月也。……外援既绝，攻益急，城中粮尽，人相食，而拒守不遗余力。（《明史·卷三百十六·列传第二百四·贵州土司》㉕*）&lt;p&gt;方官廪之告竭也，米升直二十金。食糠核草木败革皆尽，食死人肉，后乃生食人，至亲属相啖。彦方、运清部卒公屠人市肆，斤易银一两。枟尽焚书籍冠服，预戒家人，急则自尽，皆授以刀缳。城中户十万，围困三百日，仅存者千余人。（《明史·卷二百四十九·列传第一百三十七·朱燮元等》㉕）&lt;/p&gt;
#  1627年，清皇太极之天聪元年，天启七年：（清）国中大饥，斗米价银八两（天启时金一两合銀十两），人有相食者。国中银两虽多，无外贸易，是以银贱而诸物腾贵。良马银300两，牛一银百两，蟒缎一，银百五十两，布一匹，银九两。（《清太宗实录卷三》）
#  “天启辛酉，延安、庆阳、平凉旱，岁大饥。东事孔棘，有司惟顾军兴，征督如故，民不能供，道殣相望。或群职富者粟，惧捕诛，始聚为盗。盗起，饥益甚，连年赤地，斗米千钱不能得，人相食，从乱如归。饥民为贼由此而始。”&lt;ref&gt;《怀陵流寇始终录》，卷1，1页。&lt;/ref&gt;
#  1629年，崇禎二年，殺[[:w:袁崇煥|袁崇煥]]。[[:w:張岱|張岱]]《石匱書後集》：“（袁崇煥）遂於鎮撫司綁發西市，寸寸臠割之。割肉一塊，京師百姓從劊子手爭取生啖之。劊子亂撲，百姓以錢爭買其肉，頃刻立荊開腔出其腸胃，百姓群起搶之，得其一節者，和燒酒生嚙，血流齒頰間，猶唾地罵不已。拾得其骨者，以刀斧碎磔之，骨肉俱盡，止剩一首，傳視九邊。”，“时百姓怨恨，争啖其肉，皮骨已尽，心肺之间犹叫声不绝，半日而止，所谓活剐者也……百姓将银一钱，买肉一块，如手指大，噉之。食时必骂一声，须臾崇焕肉悉卖尽。”（[[:w:计六奇|计六奇]]：《[[:w:明季北略|明季北略]]》卷五）
#  1633年，崇祯六年：（陈）三接，文水人。举崇祯六年乡试，知河间县。岁旱饥，人相食。（《明史·卷二百九十一·列传第一百七十九·忠义三》㉕*）
#  1634年，崇祯七年：七年，京师饥，御史龚廷献绘《饥民图》以进。太原大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;七年，西北大旱，秦、晋人相食，（吴甘来）疏请发粟以振。（《明史·卷二百六十六·列传第一百五十四·马世奇等》㉕）&lt;/p&gt;
#  1636年，崇祯九年：山西大饥，人相食。（《明史·卷二十三·本纪第二十三·庄烈帝一》㉕*）
#   1637年，崇祯十年：十年浙江大饥，父子、兄弟、夫妻相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  崇禎十二年（1639年）[[:w:鄭鄤|鄭鄤]]以「杖母、姦妹」罪被磔死。《[[:w:明季北略|明季北略]]》记载鄭鄤被凌迟三千六百刀後，为“都人士”药用：“炮声响后，人皆跻足引领，顿高尺许，拥挤之极……归途所见，买生肉为疮疥药料者，遍长安市。二十年前之文章气节，功名显宦，竟与参术甘皮同奏肤功，亦大奇也。”
#  1639年，崇祯十二年：十二年，两畿、山东、山西、陕西、江西饥。河南大饥，人相食，卢氏、嵩、伊阳三县尤甚。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1640年，崇禎十三年，全國有123州縣發生“人相食”，98州縣蝗災。{{Citation needed}}&lt;p&gt;是年，两畿、山东、河南、山、陕旱蝗，人相食。（《明史·卷二十四·本纪第二十四·庄烈帝二》㉕*）&lt;/p&gt;&lt;p&gt;关河大旱，人相食，土寇蜂起，陕西窦开远、河南李际遇为之魁，饥民从之，所在告警。（《明史·卷二百五十二·列传第一百四十·杨嗣昌等》㉕）&lt;/p&gt;&lt;p&gt;十三年，北畿、山东、河南、陕西、山西、浙江、三吴皆饥。自淮而北至畿南，树皮食尽，发瘗胔（坟墓里的尸体）以食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕）&lt;/p&gt;
#  1641年，崇祯十四年：德州斗米千钱，父子相食，行人断绝。大盗滋矣。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;/p&gt;&lt;p&gt;及崇祯时，常洵地近属尊，朝廷尊礼之。常洵日闭阁饮醇酒，所好惟妇女倡乐。秦中流贼起，河南大旱蝗，人相食，民间藉藉，谓先帝耗天下以肥王，洛阳富于大内。（《明史·卷一百二十·列传第八·诸王五》㉕）&lt;/p&gt;&lt;p&gt;芳奕，慷慨负智略，与秉衡同举于乡，为昌乐知县。解官归，岁大歉，人相食，倾橐济之。（《明史·卷二百九十三·列传第一百八十一·忠义五》㉕）&lt;/p&gt;&lt;p&gt;十四年（左懋第）督催漕运，道中驰疏言：“臣自静海抵临清，见人民饥死者三，疫死者三，为盗者四。米石银二十四两，人死取以食，惟圣明垂念。”（《明史·卷二百七十五·列传第一百六十三·张慎言等》㉕）&lt;/p&gt; 崇禎十四年（1641年），「浙江大旱，飛蝗蔽天，食草根幾盡，人饑且疫」。崇祯十四年二月，时山东荒旱，寇盗益炽，徐德（南端到北端）数千里-{}-白骨纵横，父子相食，人迹断绝。（彭贻孙《平寇志》）
#  1641年，崇祯十四年：（九月）十一日，秦师食尽，宗龙杀马骡以享军。明日，营中马骡尽，杀贼取其尸分啖之。（《明史·卷二百六十二·列传第一百五十·傅宗龙等》㉕*）
#  明朝末年，四川大饑，“蜀大飢，人相食。先是丙戌、丁亥，連歲干涸，至是彌甚。赤地千里，糲米一斗價二十金，養麥一斗價七八金，久之亦無賣者篙芹木葉，取食殆盡。時有裹珍珠二昇，易一面不得而殆：有持數百金，買一飽不得而死。於是人皆相食，道路飢殍，剝取殆盡。無所得，父子、兄弟、夫妻，轉相賊殺。”（清·彭遵泅《蜀碧》卷四）
#  「庚辰山西大饑，人相食，剖心，其竅多寡不等。或無竅，或五六，其二、三竅為多，心大小各異。」（《[[:w:棗林雜俎|棗林雜俎]]·和集》）
#  明朝崇禎末年，河南和山東發生饑荒和蝗災，可以吃的東西都已經吃完，唯一剩下的可以吃的就只有人，於是便有了公開的人肉市場，其販賣的乃是活生生的人，稱之曰“[[:w:菜人|菜人]]”。[[:w:紀昀|紀昀]]《[[:w:閱微草堂筆記|閱微草堂筆記]]》有這樣的記載：“婦女幼孩，反接鬻於市，謂之菜人”。&lt;ref&gt;{{cite wikisource |title=《閱微草堂筆記》 |wslink=閱微草堂筆記 |chapter=卷2 |author=紀昀 | authorlink=紀昀}}&lt;/ref&gt;而在[[:w:屈大均|屈大均]]創作的一首七言古詩《[[s:菜人哀|菜人哀]]》，內容即以第一視角描述一對夫妻在崇禎末年，一位丈夫因過於飢餓，將妻子賣予一家屠戶成為“菜人”。
#  《陕西通志》第86卷载有明朝崇祯年间[[:w:马懋才|马懋才]]的《备陈灾变疏》，疏中写道：“臣乡延安府，自去岁一年无雨，草木枯焦。八、九月间，民争采山间蓬草而食，其粒类糠皮，其味苦而涩，食之仅可延以不死。至十月以后而蓬尽矣；则剥树皮而食。诸树惟榆树差善，杂他树皮以为食，亦可稍缓其死。殆年终而树皮又尽矣，则又掘山中石块而食。甘石名青叶，味腥而腻，少食辄饱，不数日则腹胀下坠而死。民有不甘于食石以死者始相聚为盗，而一、二稍有积贮之民遂为所劫，而抢掠无遗矣。有司亦不能禁治。间有获者亦恬不知畏；且曰：“死于饥与死于盗等耳，与其坐而饥死，何若为盗而死，犹得为饱鬼也。”
#  [[:w:計六奇|計六奇]]說：“天降奇荒，所以资自成也！”&lt;ref&gt;{{cite wikisource |title=《明季北略》 |wslink=明季北略 |chapter=卷05 |author=計六奇|authorlink=計六奇}}&lt;/ref&gt;。 
#  崇禎十四年（1641年）二月，[[:w:李自成|李自成]]攻陷洛陽，殺重達三百六十多斤的福王[[:w:朱常洵|朱常洵]]，用他的肉和皇家園林裡的[[:w:梅花鹿|梅花鹿]]一同烹煮，在洛陽西關周公廟舉行宴會，賜給部下食用，名曰“福祿宴”。&lt;ref&gt;《明季北略·卷十七》：王体肥，重三百余筋，贼置酒大会，以王为菹，杂鹿肉食之，号福禄酒。&lt;/ref&gt;
#  约1644年，顺治二年：（刘）泽清颇涉文艺，好吟咏。尝召客饮酒唱和。幕中蓄两猿，以名呼之即至。一日，宴其故人子，酌酒金瓯中，瓯可容三升许，呼猿捧酒跪送客。猿狰狞甚，客战掉，逡巡不敢取。泽清笑曰：“君怖耶？”命取囚扑死阶下，剜其脑及心肝，置瓯中，和酒，付猿捧之前。饮酹，颜色自若。其凶忍多此类。（《明史·卷二百七十三·列传第一百六十一·左良玉等》㉕*）
#   明末：中原盗起十余年，所在荼毒，督抚莫能办，率倡抚议，苟且幸无事，盗且服且叛。而河南比年大旱蝗，人相食，民益蜂起为盗。（《清史稿·卷五百·列传二百八十七·遗逸一》㉕*）
#  时有将军安氵侃者，一岁丧母，事其父以孝闻。父病革，刲臂为汤饮父，父良已。（《明史·卷一百十六·列传第四·诸王》㉕*）
#  襄陵王冲秌，宪王第二子，有至性。母病，刲股和药，病良已。（《明史·卷一百十八·列传第六·诸王三》㉕*）
# （襄陵王冲秌之）子范址服其教，母荆罹危疾，亦刲股进之，愈。（《明史·卷一百十八·列传第六·诸王三》㉕*）
#  刘铉，字宗器，长洲人。生弥月而孤。及长，刲股疗母疾。母卒，哀毁，以孝闻。（《明史·卷一百六十三·列传第五十一·李时勉等》㉕*）
# （孙）祖寿初守固关，遘危疾，妻张氏割臂以疗，绝饮食者七日。祖寿生，而张氏旋死，遂终身不近妇人。（《明史·卷二百七十一·列传第一百五十九·贺世贤》㉕*）
#  朱鉴，字用明，晋江人。童时刲股疗父疾。举乡试，授蒲圻教谕。（《明史·卷一百七十二·列传第六十·罗亨信等》㉕*）
#  储巏，字静夫，泰州人。九岁能属文。母疾，刲股疗之，卒不起。（《明史·卷二百八十六·列传第一百七十四·文苑二》㉕*）
#  许琰，字玉仲，吴县人。幼有至性，尝刲臂疗父疾。（《明史·卷二百九十五·列传第一百八十三·忠义七》㉕*）
#  沈德四，直隶华亭人。祖母疾，刲股疗之愈。己而祖父疾，又刲肝作汤进之，亦愈。洪武二十六年被旌。寻授太常赞礼郎。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  上元姚金玉、昌平王德儿亦以刲肝愈母疾，与德四同旌。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  至二十七年九月，山东守臣言：“日照民江伯儿，母疾，割肋肉以疗，不愈。祷岱岳神，母疾瘳，愿杀子以祀。已果瘳，竟杀其三岁儿。”帝大怒曰：“父子天伦至重。《礼》父服长子三年。今小民无知，灭伦害理，亟宜治罪。”遂逮伯儿，仗之百，遣戍海南。因命议旌表例。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  永乐间，江阴卫卒徐佛保等复以割股被旌。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  夏子孝，字以忠，桐城人。六岁失母，哀哭如成人。九岁父得危疾，祷天地，刲股六寸许，调羹以进，父食之顿愈。翌日，子孝痛创，父诘其故，始知之。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  金子良亦有孝行，父病，刲股为羹以进，旋愈。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  唐俨，全州诸生也。父荫，郴州知州，归老得危疾。俨年十二，潜割臂肉进之，疾良已。及父殁，哀毁如成人。其后游学于外，嫡母寝疾。俨妻邓氏年十八，奋曰：“吾妇人，安知汤药。昔夫子以臂肉疗吾舅，吾独不能疗吾姑哉？”于是割胁肉以进，姑疾亦愈。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  刘孝妇，新乐韩太初妻。……刘事姑谨，姑道病，刺血和药以进。……及姑疾笃，刲肉食之，少苏，逾月而卒，殡之舍侧。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  程氏，扬州胡尚絅妻。尚絅婴危疾，妇刲腕肉啖之，不能咽而卒。妇号恸不食二日。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  杨泰奴，仁和杨得安女。许嫁未行。天顺四年，母疫病不愈。泰奴三割胸肉食母，不效。一日薄幕，剖胸取肝一片，昏仆良久。及苏，以衣裹创，手和粥以进，母遂愈。母宿有膝挛疾，亦愈。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  后有张氏，仪真周祥妻。姑病，医百方不效。一方士至其门曰：“人肝可疗。”张割左胁下，得膜如絮，以手探之没腕，取肝二寸许，无少痛，作羹以进姑，病遂瘳。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  李孝妇，临武人，名中姑，适江西桂廷凤。姑邓患痰疾，将不起，妇涕泣忧悼。闻有言乳肉可疗者，心识之。一日，煮药，巘香祷灶神，自割一乳，昏仆于地，气已绝。廷凤呼药不至，出视，见血流满地，大惊呼救，倾骇城市，邑长佐皆诣其庐，命亟治。俄有僧踵门曰：“以室中蕲艾傅之，即愈。”如其言，果苏，比求僧不复见矣。乃取乳和药奉姑，姑竟获全。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  洪氏，怀宁章崇雅妻。崇雅早卒，洪守志十年。姑许，疾不能起，洪剜乳肉为羹而饮之，获愈，余肉投池中，不令人知。数日后，群鸭自水中衔出，鸣噪回翔，小童获以告姑。姑起视之，乳血犹淋漓也。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  倪氏，兴化陆鳌妻。性纯孝，舅早世，悯姑老，朝夕侍寝处，与夫睽异者十五年。姑鼻患疽垂毙，躬为吮治，不愈，乃夜焚香告天，割左臂肉以进，姑啖之愈。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  刘氏，张能信妻，太仆卿宪宠女，工部尚书九德妇也。性至孝，姑病十年，侍汤药不离侧。及病剧，举刀刲臂，侍婢惊持之。舅闻，嘱医言病不宜近腥腻，力止之。逾日，竟刲肉煮糜以进，则乃姑已不能食，乃大悔恨曰：“医绐我，使姑未鉴我心。”复刲肉寸许，恸哭奠箦前，将阖棺，取所奠置棺中曰：“妇不获复事我姑，以此肉伴姑侧，犹身事姑也。”乡人莫不称其孝。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  （颍）州又有台氏，诸生张云鹏妻。夫病，氏单衣蔬食，祷天愿代，割臂为糜以进。（《明史·卷三百三·列传第一百九十一·列女三》㉕*）

==清==
《清史稿》记载的割肉疗亲的事迹比二十五史以往各朝都多，但其实雍正有一段诏书不赞同割肉疗亲，朝廷的实际做法似乎是迫于民情不得已的情况下低调褒奖（“破格报可”），社会风气看来是称赞这种行为的。

* 雍正六年三月壬子，世宗谕曰：“……父母爱子，无所不至，若因己病而致其子割肝刲股以充饮馔、和汤药，纵其子无恙，父母未有不惊忧恻怛惨惕而不安者，况因此而伤生，岂父母所忍闻乎？父母有疾，固人子尽心竭力之时，傥能至诚纯孝，必且感天地、动鬼神，不必以惊世骇俗之为，著奇于日用伦常之外。……倘训谕之后，仍有不爱躯命，蹈于危亡者，朕亦不概加旌表，以成激烈轻生之习也。”（《清史稿·卷四百九十七·列传二百八十四·孝义一》）
* 清兴关外，俗纯朴，爱亲敬长，内悫而外严。既定鼎，礼教益备。定旌格，循明旧。亲存，奉侍竭其力；亲殁，善居丧，或庐于墓；亲远行，万里行求，或生还，或以丧归。友于兄弟，同居三五世以上，号义门，及诸义行，皆礼旌。亲病，刲股刳肝；亲丧，以身殉：皆以伤生有禁，有司以事闻，辄破格报可。所以教民者，若是其周其密也。国史承前例，撰次孝友传，亦颇及诸义行。（《清史稿·卷四百九十七·列传二百八十四·孝义一》）

历史记载中清朝的食人事件：

#  努尔哈赤时代：扬古利，舒穆禄氏，世居浑春。父郎柱，为库尔喀部长，率先附太祖，……扬古利手刃杀父者，割耳鼻生啖之，时年甫十四，太祖深异焉。（《清史稿·卷二百二十六·列传十三·扬古利等》㉕*）
#  清初：虞尔忘、尔雪，江南无锡人。国初江南多盗，尔忘、尔雪父罕卿董乡团，……罕卿死桥下矣。……知为盗杜息（所杀）。….. 比明，尔忘抱罕卿木主至，尔雪于其旁爇釜，尔忘取（杜）息舌，尔雪探心肝，且祭且啖，尔忘乃断（杜）息头。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  1627年，天聪元年，《太宗实录卷三》：“时国中大饥，斗米价银八两，人有相食者。国中银两虽多，无外贸易，是以银贱而诸物腾贵。良马，银三百两。牛一，银百两。蟒缎一，银百五十两。布匹一，银九两。盗贼繁兴，偷窃牛马，或行劫杀。于是诸臣入奏曰：盗贼若不按律严惩，恐不能止息。上恻然，谕曰：今岁国中因年饥乏食，致民不得已而为盗耳。缉获者，鞭而释之可也。遂下令，是岁谳狱，姑从宽典。仍大发帑金，散赈饥民。”
#  1631年，皇太极天聪四年：顷大凌河之役，城中人相食，明人犹死守，及援尽城降，而锦州、松、杏犹不下。（《清史稿·卷二·本纪二·太宗本纪一》㉕*）&lt;p&gt;旋有王世龙者，越城出降，言城中粮竭，商贾诸杂役多死，存者人相食，马毙殆尽。（《清史稿·卷二百三十四·列传二十一·孔有德等》㉕）&lt;/p&gt;&lt;p&gt;祖大壽疏奏：“被圍將及三月，城中食盡，殺人相食。”（《崇禎長編》卷五二）。&lt;/p&gt;&lt;p&gt;明大凌河城內，糧絕薪盡。軍士飢甚，殺其修城夫役及商賈平民為食，析骸而炊。又執軍士之羸弱者，殺而食之。（《清太宗實錄·卷十》）&lt;/p&gt;
#  1635年，皇太极天聪八年：先是，察哈尔林丹西奔图白特，其部众苦林丹暴虐，逗遛者什七八，食尽，杀人相食，屠劫不已，溃散四出。（《清史稿·卷二·本纪二·太宗本纪一》㉕*）
#  1645年，顺治二年：二年，耒（枣？）阳、襄阳、光化、宜城大饥，人相食。”（{{cite wikisource |title=《清史稿·卷44·志十九·災異五》 |wslink=清史稿/卷44 |author=趙爾巽|authorlink=趙爾巽}}㉕*）
#  1648年，顺治五年：五年春，广州、鹤庆（大理，洱海之北）嵩明（昆明市东北）大饥，人相食。”（{{cite wikisource |title=《清史稿·卷42·志十七·災異三》 |wslink=清史稿/卷42 |author=趙爾巽|authorlink=趙爾巽}}㉕*）
#  順治九年八月，漳州被圍半年，城中缺糧，一碗稀粥索價白銀四兩。居民以老鼠、麻雀、樹根、樹葉、水萍、紙張和皮革等物為食，餓死者不計其數，“城中人自相食，百姓十死其八，兵馬盡皆枵腹”&lt;ref&gt;《明清史料》丁編，第一本，第七十五頁《查明漳州解圍功次殘件》。&lt;/ref&gt;。
#  1654年，顺治十一年：顺治十一年，明将李定国攻新会，城守阅八月，食尽，杀人马为食。（《清史稿·卷五百十·列传二百九十七·列女三》㉕*）
#  顺治年間，“安邑知县鹿尽心者，得痿痺疾。有方士挟乩术，自称刘海蟾，教以食小儿脑即愈。鹿信之，辄以重价购小儿击杀食之，所杀伤甚众，而病不减。因复请于乩仙，复教以生食乃可愈。因更生凿小儿脑吸之。致死者不一，病竟不愈而死。事随彰闻，被害之家，共置方士于法。”&lt;ref&gt;[[:w:王士祯|王士祯]]：《池北偶谈·鹿尽心》&lt;/ref&gt;
#  康熙十八年（1679年），山东“终年不雨，大饥，人相食。”（乾隆《青城（即今高青）县-{}-志》卷10）
#  1681年，康熙二十年：诇知粮将罄，人相食，与诸将环而攻之。（吴）世璠众内乱，欲擒世璠以降，世璠自杀。（《清史稿·卷二百五十四·列传四十一·赉塔等》㉕*）
#  1698年，康熙三十七年春：三十七年春，平定、乐平大饥，人相食。”（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1703年，康熙四十二年：永年（邯郸东北）、东明（大名府之南部，山东曹州西）饥。秋：沛县、亳州、东阿、曲阜、蒲县（属隰州，非蒲城县）、滕县大饥。冬，汶上、沂州、莒州、兖州、东昌、郓城大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1704年，康熙四十三年：四十三年春，泰安大饥，人相食，死者枕藉。肥城，东平大饥，人相食。武定（惠民）、滨州（武定东）、商河（武定西南）、阳信（武定北）、利津、沾化饥；兖州、登州大饥，民死大半，至食屋草；昌邑、即墨、掖县、高密、膠州大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#   1785年，乾隆五十年：秋，寿光、昌乐、安丘、诸城大饥，父子相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1786年，乾隆五十一年：五十一年春，山东各府、州、县大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）&lt;p&gt;《病榻梦痕录》卷上乾隆五十一年（１７８６）条记载了苏皖鲁等地的灾情，时灾民卖妻鬻子，“流丐载道”，“尸横道路”，尸体“埋于土，辄被人刨发，刮肉而啖”。&lt;/p&gt;
#  1801，嘉庆六年: 罗思举，字天鹏，四川东乡人。……（嘉庆）六年，歼张世龙于铁溪河，……自是转战老林，饷不时至，煮马鞯，啗贼肉以追贼。……尝酒酣袒身示人，战创斑斑，为父母刲股痕凡七，其忠孝盖出天性云。（《清史稿·卷三百四十七·列传一百三十四·杨遇春等》㉕*）
#  1832年，道光十二年：夏，紫阳大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1833年，道光十三年：夏，保康、郧县、房县饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1834年，道光十四年：十四年春，归州、兴山大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1847年，道光二十七年：二十七年，南乐饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1857年，咸丰七年：七年春，肥城、东平大饥，死者枕藉；鱼台、日照、临朐亦饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1863年，同治二年，[[:w:石達開|石達開]]的軍隊為[[:w:大渡河|大渡河]]的涨水所阻，當時石部全軍已是“覓食無所得，有相殺噬人肉者”。（许亮儒遗著《擒石野史》）
#  [[:w:陈康祺|陈康祺]]《郎潜纪闻二笔》记载“同治三、四年，皖南到处食人，人肉始买三十文一斤，后增至一百二十文一斤，句容、二溧，八十文一斤，惨矣。”
#  同治三年（1864年），皖南人相食，人肉價格大漲。《曾国藩日记》同治三年四月廿二日记载：“皖南到处食人，人肉始卖三十文一斤，近闻增至百二十文一斤，句容、二溧八十文一斤。”《曾國藩日記》又記載：“[[:w:太平天国|洪楊]]之亂，[[:w:江蘇|江蘇]]人肉賣九十文一斤，漲到一百三十文錢一斤。”
#  1866年，同治五年：五年，兰州饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1867年，同治六年：五年，（穆图善）收灵州。……明年，署陕甘总督，值岁大饥，人相食。（《清史稿·卷四百五十四·列传二百四十一·刘锦棠》㉕*）
#  1868年，同治七年：七年春，即墨、孝义厅、蓝田、沔县饥。夏，泾州大饥，人相食。《清史稿·卷四十四·志十九·灾异五》㉕*）&lt;p&gt;时庆阳大饥，人相食。（《清史稿·卷四百五十四·列传二百四十一·刘锦棠等》㉕）&lt;/p&gt;&lt;p&gt; 同治七年（1868年），[[:w:定西|定西]]、[[:w:通渭|通渭]]大旱，時逢戰亂，瘟疫並起，人相食。{{Citation needed|Date=January 2025}}&lt;/p&gt;
#  1877年，光绪三年：是岁，山、陕大旱，人相食。（《清史稿·卷二十三·本纪二十三·德宗本纪一》㉕*）&lt;p&gt;丁戊奇荒是中国华北地区发生于清朝光绪元年（1875年）至四年（1878年）之间的一场罕见的特大旱灾饥荒。灾害波及山西、直隶、陕西、河南、山东、甘肃等好几个省份，“饿殍载途，白骨盈野”，饿死的人竟达一千万以上，逃亡两千万以上。随著灾情的发展，可食之物的罄尽，“人食人”的惨剧发生了。大旱的第三年（1877年）冬天，重灾区山西，到处都有人食人现象。吃人肉、卖人肉者，比比皆是。有活人吃死人肉的，还有将老人或孩子活杀吃的……无情旱魔，把灾区变成了人间地狱! 在河南，侥幸活下来的饥民大多奄奄一息，“既无可食之肉，又无割人之力”，一些气息犹存的灾民，倒地之后即为饿犬残食。{{Citation needed|Date=January 2025}}《申报》1877年12月7日载：“今岁豫省之灾，亦不减于山右，……灾黎数百万，几有易子析骸之惨”&lt;/p&gt;
#  1900年，光绪二十六年：二十六年，两宫西狩，关中大饥，人相食，（唐）锡晋醵金四十万往赈，历二州八县，艰困不少阻。（《清史稿·卷四百五十二·列传二百三十九·洪汝奎等》㉕*）
#  1910年，宣统二年十二月：是月，江、淮饥，人相食。东三省疫。（《清史稿·卷二十五·本纪二十五·宣统皇帝本纪》㉕*）
#  1911年，宣统三年：钟麟同，字建堂，山东济宁州人。威海武备学堂毕业。……宣统三年九月初九日，七十三标兵变，夜半，自北校场入城。……以手枪自击而仆，变军碎其尸，剖心啖之。上闻，有“忠骸支解，惨不忍闻”之谕，谥忠壮。（《清史稿·卷四百九十六·列传二百八十三·忠义十》㉕*）
#  光熙，本名惠熙，字亮臣。少从盛昱游，励学。钟琦遘危疾，尝刲股和药以进。（《清史稿·卷四百六十九·列传二百五十六·恩铭等》㉕*）
#  礼堂，字和贵。事亲孝。父继宏，久疟，冬月畏火，礼堂潜以身温被。居丧如礼，笑不见齿。母遘危疾，刲股合药，私祷于神，减齿以延亲寿。（《清史稿·卷四百八十一·列传二百六十八·儒林二》㉕*）
#  宋大樽，字左彝，仁和人。弱岁，刲股愈母疾，让产其弟。（《清史稿·卷四百八十五·列传二百七十二·文苑二》㉕*）
#  潘德舆，字四农，山阳人。年五六岁，母病不食，亦不食。父咯血，刲臂肉和药进，父察其色动，泣曰：“固知儿有是也！”（《清史稿·卷四百八十六·列传二百七十三·文苑三》㉕*）
#  曾艾，字虎卿，湖南新化人。尝割左臂疗父疾。（《清史稿·卷四百八十九·列传二百七十六·忠义三》㉕*）
#  陈源兖，字岱云，湖南茶陵州人。道光十八年进士，改翰林，授编修，旋授江西吉安府。先是源兖妻易氏以源兖遘疾几殆，籥天原以身代，刲臂和药饮源兖，源兖以愈，易氏旋病卒。同乡公举孝妇，请旌于朝。（《清史稿·卷四百九十·列传二百七十七·忠义四》㉕*）
#  沈瀛，字士登，江苏吴县人。尝刲臂疗母疾。（《清史稿·卷四百九十六·列传二百八十三·忠义十》㉕*）
#  李盛山，福建罗源人。母病，割肝以救，伤重，卒。巡抚常赉疏请旌，下礼部，礼部议轻生愚孝，无旌表之例。雍正六年三月壬子，世宗谕曰：“……父母爱子，无所不至，若因己病而致其子割肝刲股以充饮馔、和汤药，纵其子无恙，父母未有不惊忧恻怛惨惕而不安者，况因此而伤生，岂父母所忍闻乎？父母有疾，固人子尽心竭力之时，傥能至诚纯孝，必且感天地、动鬼神，不必以惊世骇俗之为，著奇于日用伦常之外。……倘训谕之后，仍有不爱躯命，蹈于危亡者，朕亦不概加旌表，以成激烈轻生之习也。”盛山仍予旌表。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  吕斅孚，湖南永定人。父孟卿，贫，以客授自给。母病将殆，思肉食，斅孚方七岁，贷诸屠，屠不可，泣而归。闻母呻吟，益痛，内念股肉可啗母，取厨刀砺使利，割右股四寸许，授其女弟，方五岁，令就炉火炙以进。母疾良已，孟卿归，察斅孚足微跛，得其状，与母持以哭。斅孚曰：“毋然，儿固无所苦也。”……孟卿亦尝刲股愈父病，然斅孚割股时，初不知父有是事也。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  汪灏，江南休宁人。晨、日昂、日升，其弟也。父病咯血，灏年十六，割股和药进，良愈。后数年病足，晨割股炼为末，敷治亦愈。又数年复咯血，晨复割臂以疗。更数年，疾大作，灏复割臂，勿瘳。晨病，日昂泣曰：“吾兄割臂愈父，吾不能割以愈吾兄乎？”众尼之。懵且仆，匠治棺，日升持匠斧断指，血淋漓，调药以饮晨。有司表其门曰“一门四孝友”。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  觉罗色尔岱，满洲镶红旗人，德世库七世孙也。性笃孝。年十七，父病，医不效，乃割左臂为糜以进，病稍间，旋歾。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  康熙间，以割臂疗亲旌者，有翁杜、佟良，与色尔岱同时有克什布。翁杜，满洲镶白旗人；佟良，蒙古镶黄旗人：官防御。克什布，满洲镶红旗人，官三等侍卫。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  奚缉营，字圣辉，江苏宝山人。父士本，以孝旌。缉营幼读论语，至“父母之年，不可不知”，辄陨涕簌簌，师奇之，谓真孝子子也。母病，刲臂以疗。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  张三爱，江南歙县人。为人役。事母孝，母病，不能具药物。或谓之曰：“汝欲愈母病，盍刲肝？”三爱祷于丛祠，破腹，肝堕出，以右手劙肝，得指许，左手纳于腹，束以白麻。归以肝和羹饮母，母良愈，三爱创亦合。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  杨献恒，山东益都人。父加官，与济南杨开泰有隙，……开泰计必欲杀献恒，遣其子承恩至青州谋诸吏。献恒潜知之，持铁骨朵挟刃至所居。承恩方与吏耳语，伺其出，以铁骨朵击之，仆，急拔刀断其喉，又抉其睛啖之，诣县自陈，出所藏银为证。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  刘希向，江南山阳人。……父病，希向为割股，良愈。希向年六十，病噎，其子亦割股，刀钝，肉不决，剪之，乃下，然希向竟不瘳。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  县有嫠张陈氏，家贫，刲肉以奉姑，训予田十亩助其养。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  李孔昭，字光四，蓟州人。……崇祯十五年进士，……母病，刲股疗之。（《清史稿·卷五百一·列传二百八十八·遗逸二》㉕*）
#  萧学华妻贺，湖南安化人。贺父徙陕西，学华赘其家。年余，学华归省母，贺欲与俱，父不许，贺割股肉付夫以奉姑。姑適病，学华烹肉进，病良已。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  子日焜妻李，尝刲股愈母病，事祖姑及姑孝。姑病，割臂进，病目，舐以舌，良已。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  王钜妻施，钜，萧山人；施，富阳人。姑严，小不当意，辄呵斥，施屏息不敢声。姑病反胃甚，医以为不治，施刲股和药进，病良已，姑遇施如故。钜疾作，施视疾惫，病瘵卒，姑犹不善施。钜以刲股事告，视其尸，信，乃大恸曰：“吾负孝妇！”（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  陈文世妻刘，郧人。陈、刘皆农家，刘待年于陈。既婚，姑年七十二，病噎，刘割臂和药以进，疾少间；既而复作，不食已十日，垂尽矣。刘夜屏人，杀鸡誓于神，持小刀自劙其胸二寸许，出肝刲半，取布束创，以肝与鸡同瀹汤奉姑。姑久不言，忽曰：“汤香甚！”饮之竟，病良愈，刘亦旋平。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  林经妻陈，连江人，姑盲性卞，常臆妇藐己，陈断三指自明，姑为之悔。经病，刲股；经卒，以节终。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  林云铭妻蔡，云铭，闽人；……耿精忠反，下云铭狱，蔡忧之，呕血殷紫，女瑛佩剜臂肉入药，旋苏。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  崔龙见妻钱，名孟钿，字冠之，一字浣青。龙见，永济人；钱，武进人，侍郎维城女。九岁刲臂疗父疾。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  张茂信妻方，茂信，河津人；方仪徵人。方尝割股愈舅疾，舅与茂信皆卒，奉姑刘。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  （袁）进忠病，疡生于胫，（养）女刲股以疗，家人皆不知，而长女虐愈甚。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  王前洛聘妻林，潜山人。前洛病，林父饣鬼药，林潜刲股入药。前洛卒，固请奔丧，引刀誓不嫁。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  徐文经聘妻姚，名淑金，侯官人。文经卒，淑金屡求死，乃归于徐。贫，舅殁，姑疾作，刲股以疗。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  乔涌涛聘妻方，桐城人。涌涛卒，涌涛母丁亦病，方请于父母，归于乔。以姑病寒疾，亦薄其衣当风雪。刲股以进姑，病良已。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  袁绩懋妻左，绩懋见《忠义传》。左名锡璇，字芙江，阳湖人。事亲孝，父病，刲臂和药进。工诗善画，书法尤精，著有卷葹阁诗集。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  何其仁聘妻李，路南人。嘉庆十一年，年十六，未行。其仁及其父皆病笃，李割股畀叔母使送婿家。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  林国奎妻郑，闽人。国奎卒，有子二。郑将殉，姑诫以存孤，乃已。一子殇，遂自沉于江，渔者拯以还。姑疾，刲肝杂糜进，疾良已。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  吉山妻瓜尔佳氏，名惠兴，满洲人，杭州驻防。早寡，事姑谨，尝刲肱疗姑疾。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  王如义妻向，涪州人。幼能为诗文。如义，农家子，向恒劝之读。道光十六年，如义暴卒，姑喻之嫁，矢以死。舅病，为刲股。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  许会妻张，颍州人。姑姣而虐，恶张端谨不类，日诟且挞，张事姑益恭。姑病，刲股以疗，姑虐如故。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  安于磐妻朱、后妻田，于磐，贵州蛮夷司长官。初娶朱，事姑孝，姑病，刲股，卒。复娶田，于磐病，刲股。于磐卒，抚诸子成立。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  田养民妻杨，养民，朗溪司长官；杨，邑梅司人也。年十二，母病，刲股。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  伊嵩阿，拜都氏，满洲镶黄旗人；妻希光，钮祜禄氏，正白旗人，总督爱必达女也。伊嵩阿为大学士永贵从子，早卒。方病时，希光割股进，终不起，许以死。爱必达、永贵共喻之，誓毕婚嫁乃殉。为伊嵩阿弟娶，嫁女妹及二女，次女行之明日，自缢死。张遗诗于壁，略谓：“十载要盟，此日当报命。”乾隆四十六年三月事也。永贵疏闻，高宗为赋诗，旌其节。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  朱承宇妻曹，承宇，无锡人；曹，武进人：皆农家也。生二子、一女，而承宇死。承宇弟迫之嫁，曹以死拒。……哭于承宇墓，还，遂缢。……及敛，左臂创未合，盖承宇病时尝割臂也。父为讼于县，罪迫嫁者。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）

==中华民国==
1936年“3月1日万源曹家沟某家七人，饿毙四人；余三人气息奄奄，竟为逃荒饥民杀死，分割炙食无余。”{{cfn|许汉三|y=1985}}
1936年3月19日四川省报载：“北川县人肉每斤五百文。片口镇饥民张彭氏、何张氏、陈顺氏因饥饿难忍，挖掘死尸围食，被捕。”{{cfn|许汉三|y=1985}}

1936年四川《民间意识》杂志汇载四川各地吃人的消息：“松潘半边街居民陈氏，自杀其八岁的亲生女而食，食尽仍病饿而死。沿途数百里内，人血、白骨与饿死者，填满沟壑。”{{cfn|许汉三|y=1985}}

民國30年（1941）-民國32年（1943）河南省大旱，人相食。1942年河南省赈济会推选[[:w:杨一峰|杨一峰]]、[[:w:刘庄甫|刘庄甫]]、[[:w:任兆鲁|任兆鲁]]三人等赴[[:w:重庆|重庆]]，请国民党中央免除徵賦，蒋介石拒不接见。大公报主笔[[:w:王芸生|王芸生]]在1942年的一篇《看重庆，念中原》的社论中写道：“饿死的暴骨失肉，逃亡的扶老携幼，妻离子散，挤人丛，挨棍打，未必能够得到赈济委员会的登记证。吃杂草的毒发而死，吃干树皮的忍不住刺喉绞肠之苦。把妻女驮运到遥远的人肉市场，未必能够换到几斗粮食。”[[:w:冯小刚|冯小刚]]於2012年拍摄的电影《一九四二》讲的正是这段时期发生的故事。

1948年6月[[:w:國共內戰|國共內戰]]期間，[[:w:中共|中共]]将领[[:w:林彪|林彪]]進行[[:w:長春圍城|長春圍城]]，禁止糧食進城，國軍于是收集城內的糧食，造成很多人餓死街頭。10月21日，城內守軍[[:w:鄭洞國|鄭洞國]]投降。活過來的人說，「就喝死人腦瓜殼裡的水，都是蛆。就這麼熬著，盼著，盼開卡子放人。就那麼幾步遠，就那麼瞅著，等人家一句話放生。卡子上天天宣傳，說誰有槍就放誰出去。真有有槍的，真放，交上去就放人。每天都有，都是有錢人，在城裡買了準備好的，都是手槍。咱不知道。就是知道，哪有錢買呀！」參加圍城的中共官兵說：「在外邊就聽說城裡餓死多少人，還不覺怎麼的。從死人堆裡爬出多少回了，見多了，心腸硬了，不在乎了。可進城一看那樣子就震驚了，不少人就流淚了。」&lt;ref&gt;张正隆：《雪白血红》&lt;/ref&gt;

==中華人民共和國==

=== 三年大跃进时期 ===
1959年－1961年「[[:w:大跃进|大躍進]]」期間，中國大陸發生“[[:w:三年困难时期|三年大饑荒]]”，据各方估计共造成1500万-5500万[[:w:非正常死亡|非正常死亡]]&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title=The Institutional Causes of China's Great Famine, 1959–1961|author=|url=https://www0.gsb.columbia.edu/faculty/pyared/papers/famines.pdf|first1=XIN|last2=QIAN|first2=NANCY|date=2015-01|journal=Review of Economic Studies|issue=4|doi=10.1093/restud/rdv016|others=|year=|volume=82|page=|pages=1568–1611|pmid=|last3=YARED|first3=PIERRE|archive-date=2019-09-06|url-status=|via=|last1=MENG|archive-url=https://web.archive.org/web/20190906163322/https://www0.gsb.columbia.edu/faculty/pyared/papers/famines.pdf|dead-url=no}}&lt;/ref&gt;&lt;ref name=&quot;:29&quot;&gt;{{Cite web|title=西方学术界的大跃进饥荒研究|url=http://ww2.usc.cuhk.edu.hk/PaperCollection/webmanager/wkfiles/2012/201503_38_paper.pdf|author=陈意新|date=2015-01|format=|work=[[:w:香港中文大学|香港中文大学]]|publisher=《江苏大学学报》|language=zh|archive-url=https://web.archive.org/web/20210517052743/http://ww2.usc.cuhk.edu.hk/PaperCollection/webmanager/wkfiles/2012/201503_38_paper.pdf|archive-date=2021-05-17|dead-url=no}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|title=SITES OF HORROR: MAO'S GREAT FAMINE [with Response]|author=Felix Wemheuer|url=http://www.jstor.org/stable/41262812|date=2011|journal=The China Journal|issue=66|doi=|others=|year=|editor-last=Dikötter|editor-first=Frank|volume=|page=|pages=155–164|issn=1324-9347|pmid=|archive-url=https://web.archive.org/web/20200727141524/https://www.jstor.org/stable/41262812|archive-date=2020-07-27|dead-url=no}}&lt;/ref&gt;。餓殍遍野，到處都有餓死倒斃在路邊的人，有些地方甚至出現吃人肉的現象。[[:w:楊繼繩|杨继绳]]所著的《[[:w:墓碑 (书籍)|墓碑]]》一書援引梁志遠的《關於「特種案件」的匯報——安徽亳縣人吃人見聞錄》記載指人吃人並不是個別現象：“其面積之廣，數量之多，時間之長，實屬世人罕見”{{cfn|楊繼繩|y=2008|p=274}}。

1960年春，吃人肉情況不斷發生，人肉的交易市場也隨之出現在城郊、集鎮、農民擺攤等{{cfn|楊繼繩|y=2008|p=278}}。三年大饑荒的[[:w:口述歷史|口述歷史]]《[[:w:尋找大饑荒倖存者|尋找大饑荒倖存者]]》记载了四十九起人吃人事件&lt;ref name=&quot;rfa&quot;&gt;{{Cite news|url=https://www.rfa.org/mandarin/pinglun/wenyitiandi-cite/yujie-01082014125845.html|title=为当代中国修筑一面“哭墙”--依娃《寻找大饥荒幸存者》|publisher=[[:w:自由亚洲电台|自由亚洲电台]]|date=2014-01-08|archive-url=https://web.archive.org/web/20210722001314/https://www.rfa.org/mandarin/pinglun/wenyitiandi-cite/yujie-01082014125845.html|archive-date=2020-07-22|dead-url=no|language=zh|author=余杰|authorlink=余杰}}&lt;/ref&gt;。人吃人事件在[[:w:四川|四川]]、[[:w:甘肅|甘肅]]、[[:w:青海|青海]]、[[:w:西藏|西藏]]、[[:w:陝西|陝西]]、[[:w:寧夏|寧夏]]、[[:w:河北|河北]]、[[:w:遼寧|遼寧]]皆有耳聞，幾乎遍及全國{{cfn|貝克|y=2005}}。據作家[[:w:沙青|沙青]]的[[:w:报告文学|報告文學]]記載：「有一戶農家，吃得只剩了父親和一男一女兩個孩子。一天，父親將女兒趕出門去，等女孩回家時，弟弟不見了，鍋裡浮著一層白花花油乎乎的東西，灶邊扔著一具骨頭。幾天之後，父親又往鍋裡添水，然後招呼女兒過去。女孩嚇得躲在門外大哭，哀求道：『爸爸，別吃我，我給你摟草、燒火，吃了我沒人給你做活。』」&lt;ref&gt;{{Cite web|title=依稀大地湾——大饥荒年代|url=https://boxun.com/news/gb/z_special/2004/12/200412281348.shtml?__cf_chl_jschl_tk__=pmd_cf65954eb189551663c797db8d490efde1f84d97-1626912600-0-gqNtZGzNAg2jcnBszQti|author=沙青|date=2004-12-28|publisher=[[:w:博讯|博讯]]|language=zh|archive-url=https://web.archive.org/web/20080822033646/http://www.peacehall.com/news/gb/z_special/2004/12/200412281348.shtml|archive-date=2008-08-22|dead-url=no}}&lt;/ref&gt;

* '''四川'''：《[[:w:中國大饑荒，1958-1962|中國大饑荒，1958-1962]]》引用的中國官方檔案中有吃人記載，如在[[:w:四川省|四川省]][[:w:石柱土家族自治縣|石柱土家族自治縣]]的桥头区，老妇人罗文秀是第一个开始吃人肉的人。在家人一家七口全部死去后，罗文秀把三岁女童马发慧的尸体挖出来。她把小女孩儿的肉割下来，用辣椒调味，然后蒸熟吃掉&lt;ref name=&quot;紐約時報&quot;&gt;{{cite news|url=http://cn.nytimes.com/china/20120917/c17famine/|title=記錄大饑荒人相食的慘劇|publisher=《[[:w:紐約時報|紐約時報]]》|date=2012年9月17日|archive-date=2013年10月23日|archive-url=https://web.archive.org/web/20131023013637/http://cn.nytimes.com/china/20120917/c17famine/|dead-url=no|author=DIDI KIRSTEN TATLOW|language=zh}}&lt;/ref&gt;。另一份1961年1月27日的文件，讲述了一个四川母亲用毛巾勒死了自己五岁大的儿子，“吃了四顿”。调查者王德明写道，“这样令人震惊的可怕事件远非只有这一起。”&lt;ref name=&quot;紐約時報&quot; /&gt;

* '''河南'''：1959年10月至1960年4月，[[:w:信阳事件|信陽事件]]，[[:w:商丘|商丘]]、[[:w:開封|開封]]餓得人身浮腫，吃樹皮，餓死100萬（到數百萬）人口，時諺：“人吃人，狗吃狗，老鼠餓得啃磚頭。”“信陽五里店村一個14、15歲的小女孩，将4、5歲的弟弟殺死煮了吃了，因爲父母都餓死了，只剩下這兩個孩子，女孩餓得不行，就吃弟弟。”{{cfn|楊繼繩|y=2008}} 河南省[[:w:固始县|固始縣]]官方記載有二百例人吃人事件，縣委以“破壞屍體”為名，逮捕群眾{{cfn|貝克|y=2005|p=180|url=https://books.google.com/books?id=hjpdAAAAIAAJ&amp;q=固始縣+二百}}。鹿邑、夏邑、虞城、永城等县共发现吃死人肉的情况20多起。据中央工作组魏震报告，鹿邑县从1959年10月到1960年11月，发现人吃人的事件6起。马庄公社马庄大队庞王庄18岁女子王玉娥于1960年4月19日将堂弟弟5岁的王怀郎溺死煮食，怀郎14岁的亲姐姐小朋也因饥饿吃了弟弟的肉。&lt;ref&gt;{{cite news |title=[杨继绳]《墓碑》――中国六十年代大饥荒纪实. |url=http://|publisher=第54頁 |accessdate=2022-03-23}}&lt;/ref&gt;

* '''甘肃'''：[[:w:通渭县|通渭縣]]，1958年全縣糧食實產8300多萬斤，虛報1.8億斤。人口大量死亡；有人回憶“1959年11月到臘月，死的人多。老百姓一想那事就要流淚。餓死老人家的，餓死婆娘的，日子過得糊裡糊塗。把人煮了吃，肉割來煮了吃……人甚麼也不想，甚麼也不怕，就想吃，想活。把娃娃、自己的娃娃吃下的，也有；把外面逃到村上的人殺了吃的，也有。吃下自己娃娃的，浮腫，中毒，不像人樣子。有的病死了，也有救下的。吃了娃娃心裡慘的，吃過就後悔了，自己恨自己。在村子里住不下去，沒人理他，嫌他臟。”（《50年代末大飢荒驚人記實》）
* '''青海'''：人吃人事件110多起，漢東公社楊家灘生產隊的婦女竟吃了9個小孩&lt;ref&gt;武文軍:《餓魂祭:中國六十年代饑荒考》，蘭州學刊2005年專輯，蘭州社會科學院主編，p110-110&lt;/ref&gt;。
* '''湖南'''：据余习广《吃人饿鬼：[[:w:刘家远惨杀亲子食子案|刘家远惨杀亲子食子案]]》記載，[[:w:湖南|湖南]][[:w:澧县|澧县]]如东公社男子刘家远，將自己儿子殺害後烹煮食用。刘家远也因食子而被處決&lt;ref&gt;{{cite news|title=毛泽东时代惨剧：三年大饥荒饥民十大奇吃|url=https://www.dwnews.com/%E4%B8%AD%E5%9B%BD/59674203/%E6%AF%9B%E6%B3%BD%E4%B8%9C%E6%97%B6%E4%BB%A3%E6%83%A8%E5%89%A7%E4%B8%89%E5%B9%B4%E5%A4%A7%E9%A5%A5%E8%8D%92%E9%A5%A5%E6%B0%91%E5%8D%81%E5%A4%A7%E5%A5%87%E5%90%83|publisher=[[:w:共识网|共识网]]|archive-date=2020-11-05|archive-url=https://web.archive.org/web/20201105165243/https://www.dwnews.com/%E4%B8%AD%E5%9B%BD/59674203/%E6%AF%9B%E6%B3%BD%E4%B8%9C%E6%97%B6%E4%BB%A3%E6%83%A8%E5%89%A7%E4%B8%89%E5%B9%B4%E5%A4%A7%E9%A5%A5%E8%8D%92%E9%A5%A5%E6%B0%91%E5%8D%81%E5%A4%A7%E5%A5%87%E5%90%83|dead-url=no|author=惠风（原作者：彭劲秀）|date=2014-03-11|language=zh|agency=[[:w:多維新聞|多維新聞]]}}&lt;/ref&gt;。
* '''安徽'''：作家[[:w:王立新 (1949年)|王立新]]1980年代曾赴[[:w:凤阳县|凤阳]]采访过，他在报告文学中写道：“梨园乡小岗生产队严俊冒告诉我：1960年，我们村附近有个死人塘，浮埋着许多饿死的人。为什么浮埋？饿得没力气呀，扔几锹土了事。说起来，对不起祖先，也对不起冤魂。人饿极了，什么事都干得出来。我的一位亲戚见人到死人塘割死人的腿肚子吃，她也去了。开始有点怕，后来惯了，顶黑去顶黑回。我问她：‘怎么能……？’她叹息道：‘饿极了。’”&lt;ref&gt;[[:w:李锐 (1917年)|李锐]]《大跃进亲历记》（南方出版社1999年版）&lt;/ref&gt;

=== 文化大革命时期 ===
{{main|:w:广西文革屠杀}}

[[:w:文化大革命|文化大革命]]時期（1966-1976年），[[:w:广西壮族自治区|广西壮族自治区]]除[[:w:广西文革屠杀|私刑、屠杀事件众多]]外，亦傳出多起食人事件&lt;ref name=&quot;:13&quot;&gt;{{Cite web|title=不反思“文革”的社会，就是个食人部落|url=http://history.people.com.cn/n/2013/0305/c200623-20680503.html|author=[[:w:张鸣 (学者)|张鸣]]|date=2013-03-05|format=|work=|publisher=《[[:w:中国青年报|中国青年报]]》|agency=[[:w:人民网|人民网]]|language=zh|archiveurl=https://web.archive.org/web/20200625141907/http://history.people.com.cn/n/2013/0305/c200623-20680503.html|archivedate=2020-06-25|dead-url=yes}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite web|title=我参与处理广西文革遗留问题|url=http://www.yhcqw.com/34/8938.html|accessdate=2019-11-29|author=晏乐斌|date=|format=|work=|publisher=《[[:w:炎黄春秋|炎黄春秋]]》|language=zh|archive-url=https://web.archive.org/web/20191207031844/http://www.yhcqw.com/34/8938.html|archive-date=2019-12-07|dead-url=yes}}&lt;/ref&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite web|title=广西文革中的吃人狂潮|url=http://www.cuhk.edu.hk/ics/21c/media/articles/c155-201605003.pdf|accessdate=|author=[[:w:宋永毅|宋永毅]]|date=|format=|publisher=[[:w:香港中文大学|香港中文大学]]|language=zh|archive-url=https://web.archive.org/web/20180127184237/http://www.cuhk.edu.hk/ics/21c/media/articles/c155-201605003.pdf|archive-date=2018-01-27|dead-url=no}}&lt;/ref&gt;。作家[[:w:鄭義 (作家)|鄭義]]曾在文革後赴廣西調查，于1993年出版《[[:w:红色纪念碑|红色纪念碑]]》一书，據他的統計廣西全省至少有一千人被食。紀錄片「文革廣西[[:w:武宣县|武宣縣]]紅衛兵吃人肉事件」評論称：“這些食人事件並不是因為飢荒，而是因為政治運動製造出來的仇恨心態&lt;ref&gt;{{Cite web |url=https://www.youtube.com/watch?v=vR2JhwcEM1A |title=文革廣西武宣縣紅衛兵吃人肉事件 |accessdate=2015-07-25 |archive-date=2016-03-16 |archive-url=https://web.archive.org/web/20160316105309/https://www.youtube.com/watch?v=vR2JhwcEM1A |dead-url=no }}&lt;/ref&gt;”。

其中人食人最厲害的地方之一是廣西[[:w:武宣县|武宣縣]]，官方调查发现至少38人被吃&lt;ref name=&quot;:0&quot; /&gt;，民间研究调查则发现有70余人&lt;ref name=&quot;:4&quot; /&gt;甚至上百人被吃&lt;ref name=&quot;:12&quot;&gt;{{Cite web|title=Chronology of Mass Killings during the Chinese Cultural Revolution (1966-1976)|url=https://www.sciencespo.fr/mass-violence-war-massacre-resistance/en/document/chronology-mass-killings-during-chinese-cultural-revolution-1966-1976|accessdate=|author=[[:w:宋永毅|宋永毅]]|date=2011-08-25|format=|publisher=[[:w:巴黎政治学院|巴黎政治学院]]（Sciences Po）|language=en|archive-url=https://web.archive.org/web/20190425062821/https://www.sciencespo.fr/mass-violence-war-massacre-resistance/en/document/chronology-mass-killings-during-chinese-cultural-revolution-1966-1976|archive-date=2019-04-25|dead-url=no}}&lt;/ref&gt;。武宣县“一女民兵因参与杀人坚定勇敢，且专吃男人生殖器而臭名远播，并因此入党做官，官至武宣县革委副主任。处遗时期中共中央书记处一天一个电话催问处理结果，并严厉责问：‘像这样的人，为什么还不赶快开除党籍？’但该副主任拒不承认专吃生殖器，只承认一起吃过人。最后的处理是开除党籍，撤销领导职务。现已调离武宣。”{{cfn|鄭義|y=1993|p=74-75|url=https://books.google.com/books?id=IJBxAAAAIAAJ&amp;q=武宣縣+副主任}}

== 参考文献 ==
=== 引用 ===
{{Reflist|30em}}

=== 来源 ===
{{refbegin}}
* 王永寬：《中國古代酷刑》
* [[:w:黃文雄 (作家)|黃文雄]]：《中國食人史》
* 黃粹涵：《中國食人史料鈔》
* {{cite book
|author=许汉三
|title=《黃炎培年谱》
|url=https://books.google.com/books?id=z2djAAAAIAAJ
|year=1985年
|publisher=文史资料出版社
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-26
|archive-url=https://web.archive.org/web/20210426094608/https://books.google.com/books?id=z2djAAAAIAAJ
}}
* {{cite book
|author=鄭義
|title=《紅色紀念碑》
|url=https://books.google.com/books?id=IJBxAAAAIAAJ
|year=1993年
|publisher=華視文化
|isbn=978-957-572-048-3
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-26
|archive-url=https://web.archive.org/web/20210426200250/https://books.google.com/books?id=IJBxAAAAIAAJ
}}
* {{cite book
|author=楊繼繩
|author-link=楊繼繩
|title=《墓碑——中國六十年代大饑荒紀實 上篇》
|url=https://books.google.com/books?id=GnglAQAAMAAJ
|year=2008年
|publisher=天地圖書
|isbn=978-988-211-909-3
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-19
|archive-url=https://web.archive.org/web/20210419003552/https://books.google.com/books?id=GnglAQAAMAAJ
}}
* {{cite book 
| author=賈斯柏‧貝克 
| translator=姜和平  
| title=《餓鬼：毛時代大饑荒揭秘》
| publisher=明鏡出版社 
| date=2005年10月 
| url=http://books.google.com/books?id=hjpdAAAAIAAJ 
| isbn=978-1-932138-30-6
| ref = {{SfnRef|貝克|2005}}}}
* [[:w:有線電視|有線電視]]財經資訊台《神州穿梭》 「文革廣西武宣縣紅衛兵吃人肉事件」
{{refend}}

== 外部链接 ==
*[[:w:钱理群|钱理群]]：《[http://www.aisixiang.com/data/3951-2.html 钱理群：说“食人”——周氏兄弟改造国民性思想之一]》{{Wayback|url=http://web.archive.org/web/20150605170543/http://www.aisixiang.com/data/3951-2.html |date=20150605170543 }}

[[Category:History of China]]</text>
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Among all major civilizations worldwide, China has the most recorded instances of cannibalism.&lt;ref&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社, 1994, &quot;中国封建时代的有关（食人习俗的）文字记载是极为丰富的。可以说，中国封建时代的食人习俗证据远比其他时代或其他国家为多&quot;&lt;/ref&gt; This entry documented 388 cannibalism cases recorded in 530 instances from the ''Twenty-Five Histories'' ([[w:Twenty-Four Histories|Twenty-Four Histories]] and [[w:Draft History of Qing|Draft History of Qing]]), consistent with prior research &lt;ref name=鄭麒來統計&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社，1994年版，第153-154页。&lt;/ref&gt;. According to another study, the [[w:Complete Classics Collection of Ancient China|Complete Classics Collection of Ancient China]], a comprehensive Chinese encyclopedic work, recorded 653 cases of filial piety act involving cutting own flesh to cure parents' illness&lt;ref name=鄭麒來統計/&gt;.

Several factors are generally considered responsible for this prevalence. 

* China experienced more famines than any other major civilizations.&lt;ref&gt;邓拓，《中国救荒史》，1937年，“我国灾荒之多，世界罕有，就文献可考的记载来看，从公元前十八世纪，直到公元二十世纪的今日，将近四千年间，几于无年无灾，也几乎无年无荒。西欧学者甚至称我国为‘饥荒的国度’（The Land of Famine)。” &lt;/ref&gt; 
* China experienced the most frequent and intense conflicts among major civilizations.&lt;ref&gt;秦晖，《中国历史上，何来如此深仇大恨》，“中国秦以后历代王朝的寿命不但比‘封建’时代的周‘王朝’和欧洲、日本的宗主王系（不是dynasty）短很多，其‘改朝换代’的巨大破坏性更几乎是人类历史上独有的。……世界史上别的民族有遭到外来者屠杀而种族灭绝的，有毁灭于庞贝式的自然灾变的，但像中国这样残忍的自相残杀确实难找他例。”&lt;/ref&gt; &lt;ref&gt; 福山《政治秩序的起源》，2014年，广西师范大学出版社，第7章，“与其他军事化社会相比，周朝的中国异常残暴。有个估计，秦国成功动员了其总人口的8%到20%，而古罗马共和国的仅1%，希腊提洛同盟的仅5.2%，欧洲早期现代则更低”&lt;/ref&gt; 
* Specific cultural beliefs developed in China, including:
** Rationalizing cannibalism as a means of expressing animosity&lt;ref&gt;《左传·襄公二十一年》，“然二子者，譬如禽兽，臣食其肉而寝处其皮矣”；岳飞，《满江红》，“壮志饥餐胡虏肉，笑谈渴饮匈奴血”；《三国演义》、《水浒传》多处有吃仇人肉的描写；等等&lt;/ref&gt;.
** Attributing medicinal properties to human flesh &lt;ref&gt;唐，陈藏器，《本草拾遗》；明，李时珍，《本草纲目》&lt;/ref&gt;.
** Viewing the practice of cutting own flesh to treat elder relatives as a noble demonstration of filial piety&lt;ref&gt; 《宋史· 卷四百五十六·列传第二百一十五·孝义》：“太祖、太宗以来，……刲股割肝，咸见褒赏；”&lt;/ref&gt;
* China established a comprehensive official historical record system early on, which remained functional even during periods of significant social chaos, preserving extensive historical documentation.

==Statistics==

Key-Ray Chong categorized records of cannibalism within the Twenty-Five Histories, based on their causes.&lt;ref name=&quot;鄭麒來統計&quot; /&gt;

{| class=&quot;wikitable&quot;
|-
!Historical Records!!Subtotal!!Wartime Famine!!Wartime Hatred!!Natural Disasters!!Peace-time Hatred!!Loyalty!!Filial Piety!!Taste!!Other
|-
| [[:w:Shiji|Records of the Grand Historian(''Shiji'')]]||19||6||11	|| ||2|| || || ||
|-
| [[:w:Book of Han|Book of Han]] ||25||11||1||13|| || || || ||
|-
| [[:w:Book of the Later Han|Book of the Later Han]]||26||15|| ||11 ||||||||||
|-
| [[:w:Records of the Three Kingdoms|Records of the Three Kingdoms]]||7||4|| ||3 ||||||||||
|-
| [[:w:Book of Jin|Book of Jin]]||32||16||1||13||2 ||||||||
|-
| [[:w:Book of Wei|Book of Wei]]||8||6||1||1	||||||||||
|-
| [[:w:History of the Southern Dynasties|History of the Southern Dynasties]]||18||12||3||3 ||||||||||
|-
| [[:w:History of the Northern Dynasties|History of the Northern Dynasties]]||6||3||3 ||||||||||||
|-
| [[:w:Book of Northern Qi|Book of Northern Qi]]||2||2 ||||||||||||||
|-
| [[:w:Book of Song|Book of Song]]||2||1||1 ||||||||||||
|-
| [[:w:Book of Liang|Book of Liang]]||9||5||2||2 ||||||||||
|-
| [[:w:Book of Chen|Book of Chen]]||1||1 ||||||||||||||
|-
| [[:w:Book of Sui|Book of Sui]]||8||2||3||3||||||||||
|-
| [[:w:Historical Records of the Five Dynasties|Historical Records of the Five Dynasties]]||15||10||4|| || || ||1||||
|-
| [[:w:Old History of the Five Dynasties|Old History of the Five Dynasties]]||5||3||1||1||||||||||
|-
| [[:w:History of Jin|History of Jin]]||3||||||3||||||||||
|-
| [[:w:History of Liao|History of Liao]]||1||||||1||||||||||
|-
| [[:w:History of Yuan|History of Yuan]]||46||5||1||27||||||13||||
|-
| [[:w:History of Song (book)|History of Song]]||43||4||4||14||||||20||1	||
|-
| [[:w:History of Ming|History of Ming]]||45||5||||22||||||17	||1||
|-
| [[w:Draft History of Qing|Draft History of Qing]]||76||3||||15	||||||58||||
|-
!Total!!397!!114!!36!!132!!4!!0!!109!!2	!!

|}

However, this statistics is incomplete and partially incorrect. It omitted [[:w:Book of Zhou|Book of Zhou]], [[:w:Book of Qi|Book of Southern Qi]], [[:w:Old Book of Tang|Old Book of Tang]], [[:w:New Book of Tang|New Book of Tang]] originally included in the ''Twenty-Five Histories,'' and failed to remove duplicated records in [[:w:History of Ming|History of Ming]].

In addition to previous research, Key-Ray Chong compiled 653 cases of filial piety act involving cutting one's own flesh to cure relatives in [[w:Complete Classics Collection of Ancient China|Complete Classics Collection of Ancient China]], of which 99% involved women, and 56% of these cases involved daughters-in-law cutting their own flesh for their mothers-in-law. Although this polarization may be the result of intentional selection bias, as both male and female cases of flesh-cutting to cure relatives are well documented in the ''Twenty-Five Histories.'' 

Key-Ray Chong concluded:&lt;ref&gt; [美]郑麒来（Key Ray Chong）《中国古代的食人：人吃人行为透视》，中国社会科学出版社，1994年版，第5-8页。&lt;/ref&gt;

{{Blockquote|text=Chinese practice of survival cannibalism does not significantly differ from that of other cultures; However, &quot;learned cannibalism''(習得性食人)''&quot; in China earned unique characteristics, particularly in its historical prevalence and specific motivations.

Unlike many other regions, where religion played a central role in cannibalistic rituals, Chinese practices were largely secular, often driven by two emotional extremes: '''Virtue and Affection''', including acts performed out of loyalty (尽忠), filial piety (尽孝), or deep love. '''Vengeance and Hatred''', on the other hand, are acts performed for revenge (報仇), to wash away shames (雪恥), or out of pure animosity. To give an example, During wartimes, cannibalism was frequently practiced as a symbolic and literal act of consuming the enemy, rooted in deep-seated hatred.

It is worth noting that ''learned cannibalism'' was also associated with '''culinary appreciation''' or '''medicinal therapy''' among the upper classes. Human flesh was perceived as both a food source and a potent medicine, especially valued for enhancing sexual function. For example, Li Shizhen's [[:w:Compendium of Materia Medica|Compendium of Materia Medica]] listed 35 human organs or substances used for medicinal purposes.}}
==Xia, Zhou and Shang Dynasty==
Note that early Chinese history often blends myth with oral tradition. While these records lack contemporary archeological evidence, they are also historically significant as they reflect how later generations conceptualized the origins of social norms including cannibalism.

# c. 1940 BCE, Xia Dynasty
#: '''English:''' He [Houyi] relied on his archery and neglected civil affairs... The family retainers killed and boiled him, and fed him to his sons. His sons could not bear to eat him and died at city gate.
#: '''Original:''' {{lang|zh-cn|「……（后羿）恃其射也，不修民事而淫於原獸，棄武羅、伯因、熊髡、圉而用寒浞。……羿猶不悛，將歸自田，家眾殺而亨之。以食其子；其子不忍食諸，死於窮門。」}}
#: '''Source:''' ''Zuo Zhuan'', Chapter of Duke Xiang (《左傳·襄公》)
#  Reign of [[:w:King Weng of Zhou|King Weng of Zhou]], c.1112-1050 BCE 
#: '''English:'''　According to ''Diwang Shiji''(The Century of Emperors), [King] Zhou imprisoned King Wen(of Zhou Dynasty). King Wen's eldest son, Boyi Kao, was serving as a hostage in Yin and acted as a charioteer for King Zhou. King Zhou boiled [Boyi Kao] to make a meat soup and presented it to King Wen, saying: &quot;''A true sage should not eat a soup made of his own son.''&quot; 
#: King Wen ate it. King Zhou then remarked, &quot;Who was it said the Earl of the West (King Wen) was a sage? He ate a soup made of his own son without even realizing it.&quot;
#: '''Original:''' 「《帝王世紀》云，（紂）囚文王，文王之長子曰伯邑考，質於殷，為紂御。紂烹為羮，賜文王曰：聖人當不食其子羮。文王食之，紂曰，誰謂西伯聖者，食其子羮尚不知也。」
#: '''Source:''' Justice in History, book 3, records of Yin (《史記正義·卷三·殷本紀》)
#: '''Note:''' The ''Century of Emperors''(《帝王世紀》) cited above was written in [[:w:Jing Dynasty|Jin Period]], and the original is now lost.

== Spring and Autumn / Warring States Periods ==

The [[:w:Spring and Autumn period|Spring and Autumn]] and [[:w:Warring States period|Warring States]] periods (approx. 770–221 BC) marked a significant era where cannibalism was documented under various social and political motivations. Famous Chinese idioms such as &quot;exchanging children to eat&quot; (''易子而食'', from [[:w:Zuo Zhuan|Zuo Zhuan]]) and &quot;eating the flesh and sleeping on the skin&quot; (''食肉寝皮'', from [[:w:Zuo Zhuan|Zuo Zhuan]]) both originated during this time.

Cases of cannibalism during this period can be categorized into four dominant motivations.

# '''Warfare and Siege Famines:''' The most frequent cause. During prolonged sieges, resources were so depleted that citizens resorted to &quot;exchanging children to eat&quot; to avoid consuming their own offspring.
# '''Political motivation:''' A famous case is Yi Ya (易牙), who steamed his own son to serve as a delicacy to Duke Huan of Qi to prove his absolute loyalty.
# '''Intimidation:''' Cannibalism was used as a tool of terror or vengeance. Examples include the Di people killing and eating Duke Yi of Wei(''狄人殺食衛懿公''), or the Ruler of Zhongshan boiling the son of the his own general, Yue Yang(''中山君烹樂羊子''), to test his loyalty.
# '''Cultural customs:''' Early records mention peripheral groups, such as the &quot;People-Eating Kingdom&quot; (啖人國), though these may be the result of Han-centric view of &quot;barbaric&quot; outsiders.

While the [[:w:Zuo Zhuan|Zuo Zhuan]] records at least 15 major natural famines, there are no explicit records of cannibalism resulting from &quot;natural&quot; disasters during this specific period. 

However, historians often note that the absence of such records does not necessarily prove the absence of the practice; rather, it may reflect the selective focus bias on military and political events over lower-class sufferings.

=== Before Warring State period ===

# The practice of &quot;Yi Di&quot; (''宜弟'')
#: '''English''': In the ancient past, there was a kingdom called Kaishu to the east of Yue. When a first-born son was born, they would dismember and eat him. The practice is called &quot;Yi Di&quot; (meaning &quot;benefiting the younger brothers&quot;).
#: '''Original:''' 昔者越之東有輆沭之國者，其長子生，則解而食之，謂之「宜弟」。
#: '''Source:''' ''[[:w:Mozi|Mozi(Book)]]'', &quot;Moderation in Funerals&quot; (《墨子·節葬下》)
# Critique of &quot;Yi Di&quot;, by Mozi
#: '''English:''' Luyang Wenjun said to Mozi: &quot;South of Chu, there is a kingdom of man-eaters called Qiao. When a first-born son is born, they butcher and eat him, calling it 'Yi Di.' If the meat is flavorful, they present it to their ruler, who rewards the father. Is this not a detestable custom?&quot; 
#: Mozi replied: &quot;Even the customs of the Central Kingdoms are similar. How is killing a father and rewarding his son any different from eating a son and rewarding his father? If we do not govern by Benevolence and Righteousness, how can we criticize the barbarians for eating their sons?&quot;
#: '''Original:''' {{lang|zh-tw|魯陽文君語子墨子曰：「楚之南有啖人之國者橋，其國之長子生，則鮮而食之，謂之宜弟。美，則以遺其君，君喜則賞其父。豈不惡俗哉？」子墨子曰：「雖中國之俗，亦猶是也。殺其父而賞其子，何以異食其子而賞其父者哉？苟不用仁義，何以非夷人食其子也？」}}
#: '''Source:''' ''[[:w:Mozi|Mozi(Book)]]'', &quot;Lu Wen&quot; (《墨子·魯問》)
# Ethnographic Records of the Wuhu
#: '''English:''' To the west of the Nanman (Southern Barbarians) lies the Kingdom of Man-eaters, named [[:w:Cochin|Cochin]](Crossed rivers). There, man and woman bath in the same river, thus the name.
#: It is their custom to always dismember and eat the first-born son, calling it &quot;Yi Di.&quot; If the taste is delicious, they offer it to their ruler, who in turn rewards the father. Furthermore, if a man marries a beautiful wife, he offer her to his elder brother. These people are known today as the Wuhu.
#: '''Original:''' {{lang|zh-tw|其西有啖人國，生首子輒解而食之，謂之宜弟。味旨，則以遺其君，君喜而賞其父。取妻美，則讓其兄。今烏滸人是也。}}
#: '''Source:''' ''[[:w:Book of the Later Han|Book of the Later Han]]'', &quot;On the Southern and Southwestern Barbarians&quot; (《後漢書·南蠻西南夷列傳》)

=== In Warring State period ===
# During reigns of Duke Huan of Qi (''齊桓公'', r. 685–643 BCE)
#: '''English''': During the reign of Duke Huan of Qi, Yi Ya served the Duke as his personal chef. The Duke once said that he had never tasted steamed infant. Upon hearing this, Yi Ya steamed his own firstborn son and presented the dish to the Duke. Human nature is such that one loves one's own children; yet he who does not love his own son. Then, what he would do to his own lord?
#: '''Original:''' 夫易牙以调和事（齐桓）公，公曰&quot;惟蒸婴儿之未尝&quot;，于是蒸其首子而献之公。人情非不爱其子也，于子之不爱，将何有于公？
#: '''Source:''' ''[[:w:Guanzi (text)|Guanzi]]'', &quot;Minor Exaltation&quot; (《管子·小称》)
## Alternate records of &quot;Yi Ya&quot;, During reigns of Duke Huan of Qi (''齊桓公'', r. 685–643 BCE)
##: '''English''': Duke Huan of Qi was fond of rare delicacies, and so Yi Ya steamed his own son's head and presented it to him.
##: '''Original:''' 齐桓公好味，易牙蒸其子首而进之。
##: '''Source:''' ''[[:w:Han Feizi|Han Feizi]]'', &quot;The Two Handles&quot; (《韓非子·二柄·難一》)
# 660 BCE: The Death and Consumption of Duke Yi of Wei (''衛懿公'')
#: '''English''': The Di people arrived and overtook Duke Yi of Wei at Rongze, where they killed him. They consumed all of his flesh, only his liver was untouched.
#: '''Original:''' 狄人至，及（卫）懿公于荣泽，杀之，尽食其肉，独舍其肝。
#: '''Source:''' ''[[:w:Lüshi Chunqiu|Lüshi Chunqiu]]'' (《吕氏春秋》)
# 594 BCE: The Siege of Song
#: '''English''': The people of Song, fearing for their lives, sent Hua Yuan on a secret night mission into the Chu encampment. He climbed into the bed of Zi Fan and roused him, saying: &quot;Our lord has sent me, Yuan, to convey our dire situation: our city is reduced to trading children for food and splitting bones for fuel. Even so, a covenant made beneath the city walls — one that would mean the ruin of our state — we cannot accept. Withdraw thirty li (''unit of length, approx. 3 kilometers long)'' from us, and we will obey every command.&quot;
#: '''Original:''' 宋人惧，使华元夜入楚师，登子反之床，起之曰：&quot;寡君使元以病告，曰：'敝邑易子而食，析骸以爨。虽然，城下之盟，有以国毙，不能从也。去我三十里，唯命是听。'&quot;
#: '''Source:''' ''[[:w:Zuo Zhuan|Zuo Zhuan]]'', &quot;The Fifteenth Year of Duke Xuan&quot; (《左傳·宣公十五年》)
## 594 BCE: The Siege of Song  (alternate account)
##: '''English''': In the twentieth year of his reign, King Zhuang of Chu besieged Song in retaliation for the killing of a Chu envoy. After a siege of five months, the food supply within the city was completely exhausted. The inhabitants resorted to trading children for food and burning bones for fuel. Hua Yuan of Song went out to truthfully convey the situation to King Zhuang. The King said: &quot;Truly a man of virtue!&quot; and thereupon withdrew his forces.
##: '''Original:''' 二十年，（楚）围宋，以杀楚使也。围宋五月，城中食尽，易子而食，析骨而炊。宋华元出告以情。庄王曰：&quot;君子哉！&quot;遂罢兵去。
##: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Hereditary Houses of Chu, Vol. 40&quot; (《史記·卷四十·楚世家第十》)
# c. 500 BCE: Zhi the Robber (''盜跖'')
#: '''English''': Confucius and Liuxia Ji were friends; Liuxia Ji's younger brother was named Zhi the Robber. Zhi the Robber commanded a following of nine thousand men, swept through the empire with impunity, plundering the various lords. 
#: He stormed into dwellings, stole cattle and horses, and abducted women. Driven by greed, he cast aside all bonds of kinship, disregarding his parents and siblings, and made no offerings to his ancestors.
#: Wherever his forces passed, large states fortified their walls and small states withdrew into strongholds, and all the people suffered greatly. [...] At that time, Zhi the Robber was resting his men on the southern slope of Mount Tai, mincing human livers and eating them.
#: '''Original:''' 孔子与柳下季为友，柳下季之弟名曰盗跖。盗跖从卒九千人，横行天下，侵暴诸侯；穴室枢户，驱人牛马，取人妇女；贪得忘亲，不顾父母兄弟，不祭先祖。所过之邑，大国守城，小国入保，万民苦之。……盗跖乃方休卒徒太山之阳，脍人肝而餔之。
#: '''Source:''' ''[[:w:Zhuangzi (book)|Zhuangzi]]'', &quot;Robber Zhi&quot; (《莊子·盜跖》)
# 409 BCE: Yue Yang Drinks His Son's Broth
#: '''English''': Yue Yang served as a general of Wei and led an attack on Zhongshan. His son was residing in Zhongshan at the time, and the ruler of Zhongshan had the son boiled and sent the resulting broth to Yue Yang. Yue Yang sat beneath his campaign tent and drank it, finishing the entire cup. 
#: Marquis Wen of Wei said to his advisor Du Shize: &quot;Yue Yang, for my sake, ate the flesh of his own son.&quot; Du replied: &quot;One who can eat his own son's flesh. Who would he not eat?&quot; After Yue Yang had pacified Zhongshan, Marquis Wen rewarded his achievement but harbored doubts about his character.
#: '''Original:''' 乐羊为魏将而攻中山。其子在中山，中山之君烹其子而遗之羹，乐羊坐于幕下而啜之，尽一杯。文侯谓睹师赞曰：&quot;乐羊以我之故，食其子之肉。&quot;赞对曰：&quot;其子之肉尚食之，其谁不食？&quot;乐羊既罢中山，文侯赏其功而疑其心。
#: '''Source:''' ''[[:w:Zhanguo Ce|Zhanguo Ce]]'', &quot;Stratagems of Wei I, Vol. 22&quot; (《戰國策·卷二十二·魏策一》)
# 403 BCE: The Siege of Jinyang ''(晉陽之戰'')
#: '''English''': The three states of Zhi, Wei, and Han besieged Jinyang for over a year, and then diverted the Fen River to flood the city. The floodwaters rose to within three planks' breadth of the top of the walls. Within the city, cauldrons were suspended over fires for cooking, inhabitants exchanged children to eat.
#: '''Original:''' 三国（智魏韩）攻晋阳，岁馀，引汾水灌其城，城不浸者三版。城中悬釜而炊，易子而食。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Clan of Zhao, Vol. 43&quot; (《史記·卷四十三·趙世家第十三》)
## 403 BCE: The Siege of Jinyang (alternate record)
##: '''English''': The three clans of Zhi, Wei, and Han encircled the people of Zhao at Jinyang and flooded the city; the floodwaters rose to within three planks' breadth of the top of the walls, and the inhabitants resorted to eating men and horses.
##: '''Original:''' 三家（智魏韩）以国人围（赵国晋阳）而灌之，城不浸者三版，人马相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 1 (《資治通鑑·卷一》)
# 260 BCE: The Battle of Changping (''長平之戰'')
#: '''English''': By the ninth month, the Zhao soldiers had been without food for forty-six days, and in secret they began killing and ate each other.
#: '''Original:''' 至九月，赵卒不得食四十六日，皆内阴相杀食。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Bai Qi and Wang Jian, Vol. 73&quot; (《史記·卷七十三·白起王翦列傳第十三》)
## 260 BCE: The Battle of Changping (alternate record)
##: '''English''': The Zhao army was cut off from food for forty-six days, during which they secretly killed and ate each other.
##: '''Original:''' 赵军食绝四十六日，皆内阴相杀食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 5 (《資治通鑑·卷五》)
# 257 BCE: Li Tong(''李同'')'s Appeal at the Siege of Handan
#: '''English''': Li Tong said: &quot;The people of Handan are burning bones for fuel and trading children for food. Their plight could not be more desperate. Yet in your household, hundreds of concubines and maids are clothed in fine silk, with surplus grain and meat to spare, while the common people cannot complete a garment of coarse cloth and cannot fill themselves even with dregs and husks.&quot;
#: '''Original:''' 邯郸之民，炊骨易子而食，可谓急矣，而君之後宫以百数，婢妾被绮縠，馀粱肉，而民褐衣不完，糟糠不厌。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Lord Pingyuan and Yu Qing, Vol. 76&quot; (《史記·卷七十六·平原君虞卿列傳第十六》)
# c. 250 BCE: The Siege of Liaocheng
#: '''English''': Qi's general Tian Dan besieged Liaocheng for over a year, with heavy casualties among his troops, yet the city did not fall. Lu Zhonglian then composed a letter, tied it to an arrow, and shot it into the city, addressed to the Yan commander. The letter read: &quot;[...] Now you hold the exhausted people of Liaocheng against the full force of Qi's army — this is the defensive resolve of Mozi. Your men eat others and burn their bones for fuel, yet none harbor thoughts of surrender — this is the military discipline of Sun Bin. Your name shall be known throughout the realm.&quot;
#: '''Original:''' 齐田单攻聊城岁馀，士卒多死而聊城不下。鲁连乃为书，约之矢以射城中，遗燕将。书曰：……今公又以敝聊之民距全齐之兵，是墨翟之守也。食人炊骨，士无反外之心，是孙膑之兵也。能见於天下。
#: '''Source:''' ''[[:w:Shiji|Records of the Great Historian(Shiji)]]'', &quot;Biographies of Lu Zhonglian and Zou Yang, Vol. 83&quot; (《史記·卷八十三·魯仲連鄒陽列傳第二十三》)

==Han Dynasty==
The wars between the Qin and Han dynasties caused large-scale famine and population decline across China, a pattern that would recur with nearly every subsequent dynastic transition.

# Early Han Dynasty: Famine and Cannibalism Following the Collapse of Qin
#: '''English''': At the founding of the Han dynasty, inheriting the devastation left by Qin, the various lords rose simultaneously in conflict. The people abandoned their livelihoods, and a great famine ensued. Price of one shi of rice reached five thousand coins; people ate each other, more than half the population perished. Emperor Gaozu then issued an order permitting the people to sell their children, and directed the starving to seek food in Shu and Han.
#: '''Original:''' 汉兴，接秦之敝，诸侯并起，民失作业而大饥馑。凡米石五千，人相食，死者过半。高祖乃令民得卖子，就食蜀、汉。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 205 BCE: Great Famine in Guanzhong, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': A great famine struck Guanzhong; the price of one hu of rice reached ten thousand coins, and people ate each other. The people were directed to seek food in Shu and Han.
#: '''Original:''' 关中大饥，米斛万钱，人相食。令民就食蜀、汉。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Gao, Vol. 1a&quot; (《漢書·卷一上·高帝紀第一上》)
## 205 BCE: Great Famine in Guanzhong, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': A great famine struck Guanzhong; the price of one hu of rice reached ten thousand coins, and people ate each other.
##: '''Original:''' 关中大饥，米斛万钱，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 9 (《資治通鑑·卷九》)
# 196 BCE: Minced flesh of Peng Yue, in ''[[:w:Records of the Grand Historian|Shiji]]''
#: '''English''': In the eleventh year, Empress Gao put to death the Marquis of Huaiyin; (Ying) Bu grew fearful at heart. In summer, Han executed Liang Wang Peng Yue, minced his flesh into paste, and sent portions of his flesh to all the lords. 
#:When it reached Huainan, the King of Huainan was out hunting; upon beholding the paste, he trembled greatly, and secretly ordered men to muster troops, watching for signs of trouble in the neighboring commanderies.
#: '''Original:''' 十一年，高后诛淮阴侯，布因心恐。夏，汉诛梁王彭越，醢之，盛其醢遍赐诸侯。至淮南，淮南王方猎，见醢，因大恐，阴令人部聚兵，候伺旁郡警急。
#: '''Source:''' ''[[:w:Records of the Grand Historian|Shiji]]'', &quot;Biography of Qing Bu&quot; (《史记·卷九十一·黥布列传第十三》)
# 138 BCE: Flood and Famine on the Yellow River Plain, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the third spring of that year, the Yellow River overflowed onto the Pingyuan plain. Great famine, people ate each other.
#: '''Original:''' 三年春，河水溢于平原，大饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Wu, Vol. 6&quot; (《漢書·卷六·武帝紀第六》)
## 138 BCE: Flood and Famine on the Yellow River Plain, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': The Yellow River overflowed onto the Pingyuan plain. Great famine, people ate each other.
##: '''Original:''' 河水溢于平原。大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 17 (《資治通鑑·卷十七》)
# 135 BCE: Ji An's Report on Famine in Henei, ''[[:w:Shiji|Shiji]]''
#: '''English''': Ji An returned and reported: &quot;A household fire has spread to neighboring houses. it is not worth undue concern. On my way, I passed through Henan, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another. I therefore took it upon myself, acting on temporary authority, to use the imperial tally to open the Henan granaries and relieve the destitute people. I now request to return the tally and submit to punishment for acting beyond my authority.&quot; The Emperor, recognizing his virtue, pardoned him.
#: '''Original:''' 还报曰：&quot;家人失火，屋比延烧，不足忧也。臣过河南，河南贫人伤水旱万余家，或父子相食，臣谨以便宜，持节发河南仓粟以振贫民。臣请归节，伏矫制之罪。&quot;上贤而释之。
#: '''Source:''' ''[[:w:Shiji|Shiji]]'', &quot;Biographies of Ji An and Zheng Dangshi, Vol. 120&quot; (《史記·卷一百二十·汲鄭列傳第六十》)
## 135 BCE: Ji An's Report on Famine in Henei, ''[[:w:Book of Han|Book of Han]]''
##: '''English''': [Ji An] returned and reported: &quot;A household fire has spread to neighboring houses — it is not worth undue concern. On my way, I passed through Henei, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another. I therefore took it upon myself, acting on temporary authority, to use the imperial tally to open the Henei granaries and relieve the destitute people. I request to return the tally and submit to punishment for acting beyond my authority.&quot; The Emperor, recognizing his virtue, pardoned him and transferred him to serve as Prefect of Xingyang.
##: '''Original:''' 还报曰：&quot;家人失火，屋比延烧，不足忧。臣过河内，河内贫人伤水旱万余家，或父子相食，臣谨以便宜，持节发河内仓粟以振贫民。请归节，伏矫制罚。&quot;上贤而释之，迁为荥阳令。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Zhang, Feng, Ji, and Zheng, Vol. 50&quot; (《漢書·卷五十·張馮汲鄭傳第二十》)
## 135 BCE: Ji An's Report, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': On my way, I passed through Henan, where more than ten thousand families among the poor had been afflicted by flood and drought; in some cases, fathers and sons were eating one another.
##: '''Original:''' 臣过河南，河南贫人伤水旱万馀家，或父子相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 17 (《資治通鑑·卷十七》)
# 114 BCE: Famine in Shandong, ''[[:w:Shiji|Shiji]]''
#: '''English''': At that time, the eastern provinces had suffered from Yellow River floods, and for several consecutive years the harvests had failed. In some places, spanning one to two thousand li, people resorted to eating one another.
#: '''Original:''' 是时山东被河灾，及岁不登数年，人或相食，方一二千里。
#: '''Source:''' ''[[:w:Shiji|Shiji]]'', &quot;Treatise on Equalization, Vol. 30&quot; (《史記·卷三十·平準書第八》)
## 114 BCE: Famine in Shandong(the East), ''[[:w:Book of Han|Book of Han]](1)''
##: '''English''': At that time, the eastern provinces had suffered from Yellow River floods, and for several consecutive years the harvests had failed. In some places, spanning two to three thousand li, people resorted to eating one another. The Emperor, moved by compassion, ordered the famine victims to travel and seek food in the Yangtze and Huai River regions, and those who wished to remain were permitted to settle there. Imperial envoys with carriages and canopies followed one another on the roads to escort them, and grain from Ba and Shu was dispatched to provide relief.
##: '''Original:''' 是时山东被河灾，乃岁不登数年，人或相食，方二三千里。天子怜之，令饥民得流就食江、淮间，欲留，留处。使者冠盖相属于道护之，下巴、蜀粟以赈焉。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24b&quot; (《漢書·卷二十四下·食貨志第四下》)
## 114 BCE: Famine in the East, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In the third month of the third Yuanding year, water froze; in the fourth month, snow fell. In more than ten commanderies east of the passes, people ate each other.
##: '''Original:''' 元鼎三年三月水冰，四月雨雪，关东十余郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on the Five Elements, Vol. 27&quot; (《漢書·卷二十七中之下·五行志第七中之下》)
## 114 BCE: Famine in the East, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': More than forty commanderies and kingdoms east of the passes suffered famine, people ate each other.
##: '''Original:''' 关东郡、国四十馀饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 20 (《資治通鑑·卷二十》)
# 113 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In summer, the fourth month, hail fell. In more than ten commanderies and kingdoms east of the passes, Great Famine; people ate each other.
#: '''Original:''' 夏四月，雨雹，关东郡国十余饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Wu, Vol. 6&quot; (《漢書·卷六·武帝紀第六》)
# 141–87 BCE: Critique of Emperor Wu's Reign, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': &quot;Though Emperor Wu had merit in driving back the four barbarians and expanding the realm, yet he slew great numbers of his men, exhausted the people's wealth, indulged in extravagance without measure. 
#: The realm was left hollow and depleted, the hundred folk scattered and adrift, half perished. Locusts rose in great swarms, scorching the earth for thousands of li; in some places people ate each other, and the stores have not recovered to this day. 
#: He bestowed no virtue nor grace upon the people, and ought not to have temple rites established in his honour.&quot;
#: '''Original:''' 武帝虽有攘四夷广土斥境之功，然多杀士众，竭民财力，奢泰亡度，天下虚耗，百姓流离，物故者半。蝗虫大起，赤地数千里，或人民相食，畜积至今未复。亡德泽于民，不宜为立庙乐。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Sui, Liang, Xiahou, Jing, Yi and Li, Vol. 75&quot; (《漢書·卷七十五·眭兩夏侯京翼李傳第四十五》)
# c. 104 BCE: Depletion of the Realm After Dong Zhongshu, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': After Zhongshu's death, expenditures grew ever greater, the realm was hollow and depleted, and once more people ate each other.
#: '''Original:''' 仲舒死后，功费愈甚，天下虚耗，人复相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In the ninth month, eleven commanderies and kingdoms east of the passes suffered great floods. Famine; in some places people ate each other. Neighboring commanderies were called upon to render aid in coin and grain.
#: '''Original:''' 九月，关东郡国十一大水，饥，或人相食，转旁郡钱、谷''(穀)''以相救。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Yuan, Vol. 9&quot; (《漢書·卷九·元帝紀第九》)
## 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In the first year of Chuyuan under Emperor Yuan, [...] in the fifth month the Bohai Sea overflowed greatly. In the sixth month, Great Famine struck the east; many among the people starved to death, and in Langye Commandery people ate each other.
##: '''Original:''' 元帝初元元年，……其五月，勃海水大溢。六月，关东大饥，民多饿死，琅邪郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Astronomy, Vol. 26&quot; (《漢書·卷二十六·天文志第六》)
## 48 BCE: Great Famine in Eastern Commanderies, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': In autumn, the ninth month, eleven commanderies and kingdoms east of the passes suffered great floods and famine; in some places people ate each other.
##: '''Original:''' 秋，九月，关东郡、国十一大水，饥，或人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 28 (《資治通鑑·卷二十八》)
# 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In the sixth month, famine struck the east; in the land of Qi, people ate each other.
#: '''Original:''' 六月，关东饥，齐地人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Annals of Emperor Yuan, Vol. 9&quot; (《漢書·卷九·元帝紀第九》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': When Emperor Yuan ascended the throne, great floods struck the realm; eleven eastern commanderies suffered most grievously. In the second year, famine struck the land of Qi; grain reached three hundred coins per shi, many among the people starved to death, and in Langye Commandery people ate each other.
##: '''Original:''' 元帝即位，天下大水，关东郡十一尤甚。二年，齐地饥，谷''(穀)''石三百余，民多饿死，琅邪郡人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](3)''
##: '''English''': The following year, in the second month, on the day wuwu, the earth shook. That summer, in the land of Liu, people ate each other. [...] Yi Feng memorialized: &quot;The eastern lands have suffered famine for years running, compounded by pestilence; the hundred folk are wan with hunger, and some have come to eat each other. The earth trembles repeatedly, the heavens are turbid, and the light of the sun grows dim.&quot;
##: '''Original:''' 明年二月戊午，地震。其夏，刘地人相食。……（翼奉）上疏曰：……今东方连年饥馑，加之以疾疫，百姓菜色，或至相食。地比震动，天气溷浊，日光侵夺。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Sui, Liang, Xiahou, Jing, Yi and Li, Vol. 75&quot; (《漢書·卷七十五·眭兩夏侯京翼李傳第四十五》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](4)''
##: '''English''': When Emperor Yuan first ascended the throne, he summoned Yu to serve as Remonstrant Counsellor and repeatedly sought his counsel on affairs of governance. At that time the harvests had failed and many commanderies were in distress. 
##: Yu exclaimed: &quot;Now the people die of Great Famine; the dead go unburied and are eaten by dogs and swine. People eat each other, whilst the horses in the imperial stables feed on grain and grow so fat and vigorous that they must be walked daily to work it off. Is this what it means for a sovereign, having received the Mandate of Heaven, to be father and mother to the people?&quot;
##: '''Original:''' 元帝初即位，征禹為諫大夫，數虛己問以政事。是時，年歲不登，郡國多困，禹奏言：[……] 今民大飢而死，死又不葬，為犬豬食。人至相食，而廄馬食粟，苦其大肥，氣甚怒至，乃日步作之。王者受命於天，為民父母，固當若此乎！（
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Wang, Gong, Liang Gong and Bao, Vol. 72&quot; (《漢書·卷七十二·王貢兩龔鮑傳第四十二》)
## 47 BCE: Famine in Qi, ''[[:w:Book of Han|Book of Han]](5)''
##: '''English''': Kuang Heng memorialized: &quot;The eastern lands have suffered famine for years running; the hundred folk are in want and distress, and some have come to eat each other. This hath all arisen from levies and taxes being too heavy, the burdens borne by the people being too great, and the officials failing in their duty to settle and succour them.&quot;
##: '''Original:''' 匡）衡上疏曰：……今关东连年饥馑，百姓乏困，或至相食，此皆生于赋敛多，民所共者大，而吏安集之不称之效也。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Kuang, Zhang, Kong and Ma, Vol. 81&quot; (《漢書·卷八十一·匡張孔馬傳第五十一》)
## 47 BCE: Famine in Qi, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Famine struck the east; in the land of Qi, people ate each other.
##: '''Original:''' 关东饥，齐地人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 28 (《資治通鑑·卷二十八》)
# 17 BCE: Emperor Cheng's Edict Dismissing Xue Xuan, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': Emperor Cheng decreed the dismissal of Xue Xuan, saying: &quot;I, being unenlightened, have seen repeated ill omens; the harvests have failed year upon year, the granaries stand empty, the hundred folk suffer Great Famine, wandering and scattered upon the roads. Those who have perished of pestilence number in the tens of thousands; people eat each other, bandits rise on all sides, and the offices of governance lie neglected. This is owing to mine own want of virtue and the failings of mine own ministers.&quot;
#: '''Original:''' 朕既不明，变异数见，岁比不登，仓廪空虚，百姓饥馑，流离道路，疾疫死者以万数，人至相食，盗贼并兴，群职旷废，是朕之不德而股肱不良也。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biographies of Xue Xuan and Zhu Bo, Vol. 83&quot; (《漢書·卷八十三·薛宣朱博傳第五十三》)
# 15 BCE: Floods in Liang and Pingyuan, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the second year of Yongshi, the kingdoms of Liang and Pingyuan suffered floods in consecutive years; people ate each other. The regional inspectors, prefects and chancellors were held accountable and dismissed.
#: '''Original:''' 永始二年，梁国、平原郡比年伤水灾，人相食，刺史、守、相坐免。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
# 14 CE: Great Famine Along the Frontier, ''[[:w:Book of Han|Book of Han]]''
#: '''English''': In the first year of Tianfeng under Wang Mang, Great Famine struck the borderlands; people ate each other.
#: '''Original:''' 缘边大饥，人相食。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99b&quot; (《漢書·卷九十九中·王莽傳第六十九中》)
## 14 CE: Great Famine Along the Frontier, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Great Famine struck the borderlands; people ate each other.
##: '''Original:''' 缘边大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 37 (《資治通鑑·卷三十七》)
# 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](1)''
#: '''English''': In his final years, bandits rose in great numbers; armies were dispatched to suppress them, and their officers ran amok beyond the passes. In the northern borderlands and in the lands of Qing and Xu, people ate each other; east of Luoyang, grain reached two thousand coins per shi.
#: '''Original:''' 末年，盗贼群起，发军击之，将吏放纵于外。北边及青、徐地人相食，雒阳以东米石二千。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24a&quot; (《漢書·卷二十四上·食貨志第四上》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': [...] battle and slaughter, captivity by the four border peoples, criminal penalties, Great Famine, pestilence, and people eating each other had together reduced the households of the realm by half.
##: '''Original:''' 战斗死亡，缘边四夷所系虏，陷罪，饥疫，人相食，及莽未诛，而天下户口减半矣。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Treatise on Food and Commerce, Vol. 24b&quot; (《漢書·卷二十四下·食貨志第四下》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Book of Han|Book of Han]](2)''
##: '''English''': In that month, the Red Eyebrows slew the Grand Preceptor Xi Zhong Jing Shang. East of the passes, people ate each other.
##: '''Original:''' 是月，赤眉杀太师牺仲景尚。关东人相食。
##: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳第六十九下》)
## 22 CE: Collapse of Wang Mang's Realm, ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': East of the passes, people ate each other.
##: '''Original:''' 关东人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 38 (《資治通鑑·卷三十八》)
# 23 CE: The Fate of Wang Mang's Corpse, ''Book of Han''
#: '''English''': Wang Mang's severed head was sent to Gengshi and hung in the market of Wan. The common folk vied to strike and beat it; some cut out his tongue and ate it.
#: '''Original:''' 传（王）莽首诣更始，县宛市，百姓共提击之，或切食其舌。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳》)
# 23 CE: Siege of Wan — Cen Peng's Surrender, ''Hou Han Shu''
#: '''English''': When Zhen Fu fell and Cen Peng was wounded, he fled back to Wan and held the city together with Yan Shuo. Han forces besieged them for several months; the city's provisions were exhausted and people ate each other. Peng and Shuo thereupon surrendered the city.
#: '''Original:''' 汉兵攻之数月，城中粮尽，人相食，彭乃与说举城降。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Feng, Cen and Jia, Vol. 17&quot; (《後漢書·卷十七·馮岑賈列傳第七》)
# 23 CE: Siege of Wan — Cen Peng's Surrender, ''Zizhi Tongjian''
#: '''English''': [...] Han forces besieged them for several months. People within the city ate each other; they thereupon surrendered.
#: '''Original:''' 汉兵攻之数月，城中人相食，乃举城降。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 39 (《資治通鑑·卷三十九》)
# 24 CE: Li Xiong's Counsel to Gongsun Shu, ''Hou Han Shu''
#: '''English''': [...] &quot;Now the lands east of the mountains suffer Great Famine; the common folk eat each other. Where armies have passed, cities and towns are left as mounds of rubble.&quot;
#: '''Original:''' 今山东饥馑，人庶相食；兵所屠灭，城邑丘墟。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Wei Xiao and Gongsun Shu, Vol. 13&quot; (《後漢書·卷十三·隗囂公孫述列傳第三》)
# 25 CE: The Red Eyebrows Sack Chang'an, ''Book of Han''
#: '''English''': The Red Eyebrows burned the palaces and markets of Chang'an and slew Gengshi. The starving people ate each other; those who perished numbered in the hundreds of thousands. Chang'an was left a wasteland, and none walked its streets.
#: '''Original:''' 赤眉遂烧长安宫室市里，害更始。民饥饿相食，死者数十万，长安为虚，城中无人行。
#: '''Source:''' ''[[:w:Book of Han|Book of Han]]'', &quot;Biography of Wang Mang, Vol. 99c&quot; (《漢書·卷九十九下·王莽傳第六十九下》)
# 26 CE: Famine in Guanzhong, ''Hou Han Shu(1)''
#: '''English''': Great Famine struck Guanzhong; people ate each other.
#: '''Original:''' 关中饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Guangwu, Vol. 1a&quot; (《後漢書·卷一上·光武帝紀第一上》)
## 26 CE: Famine in Guanzhong, ''Hou Han Shu(2)''
##: '''English''': At that time, the three adjuncts were in great turmoil; people ate each other, the cities and towns were emptied, white bones lay strewn across the fields, and the survivors gathered here and there in fortified encampments, each holding firm.
##: '''Original:''' 时三辅大乱，人相食，城郭皆空，白骨蔽野，遗人往往聚为营保，各坚守不下。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Xuan and Liu Penzi, Vol. 11&quot; (《後漢書·卷十一·劉玄劉盆子列傳第一》)
## 26 CE: Famine in Guanzhong, ''Zizhi Tongjian''
##: '''English''': Great Famine struck the three adjuncts; people ate each other, the cities and towns were emptied, and white bones lay strewn across the fields.
##: '''Original:''' 三辅大饥，人相食，城郭皆空，白骨蔽野。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 40 (《資治通鑑·卷四十》)
# 27 CE: Siege of Ji, Zizhi Tongjian
#: '''English''': Within Zhu Fu's city of Ji, provisions were exhausted; people ate each other.
#: '''Original:''' 浮城中粮尽，人相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 41 (《資治通鑑·卷四十一》)
## 27 CE: Siege of Ji'', Hou Han Shu''
##: '''English''': Within Fu's city, provisions were exhausted; people ate each other.
##: '''Original:''' 浮城中粮尽，人相食。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Zhu, Feng, Yu, Zheng and Zhou, Vol. 33&quot; (《後漢書·卷三十三·朱馮虞鄭周列傳第二十三》)
# 27 CE: Yan Cen's Retreat to Nanyang, ''Hou Han Shu''
#: '''English''': At that time the people suffered Great Famine and ate each other; one jin of gold could be exchanged for but five sheng of beans. The roads were cut off and supplies could not get through; the soldiers subsisted on wild fruit.
#: '''Original:''' 时，百姓饥饿，人相食，黄金一斤易豆五升。道路断隔，委输不至，军士委以果实为粮。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Feng, Cen and Jia, Vol. 17&quot; (《後漢書·卷十七·馮岑賈列傳第七》)
# 109 CE: Great Famine in the Capital, ''Hou Han Shu''
#: '''English''': In the third month, Great Famine struck the capital; people ate each other.
#: '''Original:''' 三月，京师大饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor An, Vol. 5&quot; (《後漢書·卷五·孝安帝紀第五》)
## 109 CE: Great Famine in the Capital, ''Zizhi Tongjian''
##: '''English''': In the third month, Great Famine struck the capital; people ate each other.
##: '''Original:''' 三月，京师大饥，民相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 49 (《資治通鑑·卷四十九》)
# 109 CE: Floods and Famine Across the Realm, ''Hou Han Shu(1)''
#: '''English''': That year, the capital and forty-one commanderies and kingdoms suffered hail. Great Famine struck Bing and Liang; people ate each other.
#: '''Original:''' 是岁，京师及郡国四十一雨水雹。并、凉二州大饥，人相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor An, Vol. 5&quot; (《後漢書·卷五·孝安帝紀第五》)
## 109 CE: Floods and Famine Across the Realm, ''jin Shu''
##: '''English''': In the third year of Yongchu under Emperor An, floods and drought struck the realm; people ate each other.
##: '''Original:''' 安帝永初三年，天下水旱，人民相食。
##: '''Source:''' ''[[:w:Jin Shu|Jin Shu]]'', &quot;Treatise on Food and Commerce, Vol. 26&quot; (《晉書·卷二十六·志第十六·食貨》)
## 109 CE: Floods and Famine Across the Realm, ''Zizhi Tongjian''
##: '''English''': The capital and forty-one commanderies suffered floods; Great Famine struck Bing and Liang; people ate each other.
##: '''Original:''' 京师及郡国四十一雨水，并、凉二州大饥，人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 49 (《資治通鑑·卷四十九》)
# 151 CE: Drought and Famine, ''Hou Han Shu''
#: '''English''': Drought struck the capital. Great Famine afflicted Rencheng and Liang; people ate each other.
#: '''Original:''' 京师旱。任城、梁国饥，民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Huan, Vol. 7&quot; (《後漢書·卷七·孝桓帝紀第七》)
## 151 CE: Drought and Famine, ''Zizhi Tongjian''
##: '''English''': Drought struck the capital; Great Famine afflicted Rencheng and Liang; people ate each other.
##: '''Original:''' 京师旱，任城、梁国饥，民相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 53 (《資治通鑑·卷五十三》)
# 155 CE: Famine in Sili and Jizhou, ''Hou Han Shu''
#: '''English''': In the second month, famine struck Sili and Jizhou; people ate each other.
#: '''Original:''' 二月，司隶、冀州饥，人相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Huan, Vol. 7&quot; (《後漢書·卷七·孝桓帝紀第七》)
# 155 CE: Famine in Sili and Jizhou, ''Zizhi Tongjian''
#: '''English''': In the second month, famine struck Sili and Jizhou; people ate each other.
#: '''Original:''' 二月，司隶、冀州饥，人相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 53 (《資治通鑑·卷五十三》)
# 170 CE: Spousal Cannibalism in Henei and Henan, ''Hou Han Shu''
#: '''English''': In the first month of spring in the third year of Jianning, in Henei wives ate their husbands, and in Henan husbands ate their wives.
#: '''Original:''' 三年春正月，河内人妇食夫，河南人夫食妇。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Ling, Vol. 8&quot; (《後漢書·卷八·孝靈帝紀第八》)
# 194 CE: Great Drought in the Three Adjuncts, ''Hou Han Shu''
#: '''English''': A great drought struck the three adjuncts from the fourth month to this day. At that time one hu of grain fetched fifty thousand coins, and one hu of beans or wheat twenty thousand. People ate each other; white bones lay heaped in piles.
#: '''Original:''' 三辅大旱，自四月至于是月。是时谷一斛五十万，豆麦一斛二十万，人相食啖，白骨委积。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Xian, Vol. 9&quot; (《後漢書·卷九·孝獻帝紀第九》)
## 194 CE: Great Drought in the Three Adjuncts, ''Zizhi Tongjian''
##: '''English''': From the fourth month no rain fell. One hu of grain was worth fifty thousand coins; within Chang'an, people ate each other.
##: '''Original:''' 自四月不雨至于是月，谷一斛直钱五十万，长安中人相食。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# Liu Ping Spared by Cannibals, ''Hou Han Shu''
#: '''English''': Liu Ping, styled Gongzi, was a man of Pengcheng in Chu. During the upheavals of the Gengshi era, he and his mother hid together in the wilderness. 
#: One morning he went out to forage for food and was seized by starving bandits who meant to boil and eat him. He knelt and said: &quot;This morning I went to gather herbs for my aged mother, who depends on me for her life. I beg ye to let me return, feed my mother, and then come back to die.&quot; Tears streamed down his face. 
#: The bandits, moved by his sincerity, took pity and released him. Liu Ping returned, fed his mother, and then told her: &quot;I made a pledge to the bandits; honour forbids me to deceive them.&quot; He went back to the bandits. They were all greatly astonished and said to one another: &quot;We have long heard of men of fierce integrity — now we behold one. Go, friend; we have not the heart to eat thee.&quot; And so he was spared.
#: '''Original:''' 刘平字公子，楚郡彭城人也。[…] 更始时，天下乱，[…] 与母俱匿野泽中。平朝出求食，逢饿贼，将亨（通“烹”）之，平叩头曰：“今旦为老母求菜，老母待旷为命，愿得先归，食母毕，还就死。”因涕泣。贼见其至诚，哀而遣之。平还，既食母讫，因白曰：“属与贼期，义不可欺。”遂还诣贼。众皆大惊，相谓曰：“常闻烈士，乃今见之。子去矣，吾不忍食子。”于是得全。（《后汉书·卷三十九·刘赵淳于江刘周赵列传第二十九》）
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Zhao Xiao Offers Himself to Cannibals, ''Hou Han Shu''
#: '''English''': [After the fall of Wang Mang] the realm fell into turmoil and people ate each other. [Zhao Xiao's] younger brother Li was seized by starving bandits. 
#: Upon hearing this, Zhao Xiao bound himself and went to the bandits, saying: &quot;Li hath long been starved and is thin and gaunt; I filleth ye hunger better than him&quot; The bandits were greatly astonished and released them both, saying: &quot;Go home for now, and bring back rice and dried provisions instead.&quot; 
#: Xiao sought provisions but could find none; he returned to the bandits and offered himself for the pot. The bandits, marvelling at him, did him no harm.
#: '''Original:''' （王莽之後）天下乱，人相食。孝弟礼为饿贼所得，孝闻之，即自缚诣贼，曰：&quot;礼久饿羸瘦，不如孝肥饱。&quot;贼大惊，并放之，谓曰：&quot;可且归，更持米糒来。&quot;孝求不能得，复往报贼，愿就亨。众异之，遂不害。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Wang Lin Guards His Parents' Tomb, ''Hou Han Shu''
#: '''English''': In Runan there was a man named Wang Lin, a junior official, who lost his parents when he was but ten years of age. 
#: When great turmoil broke out and the people fled, only Wang Lin and his brothers remained to guard the burial mound, their weeping unceasing. His younger brother Ji went out and was seized by the Red Eyebrows, who meant to eat him. Wang Lin bound himself and begged to die in his brother's stead. 
#: The bandits, moved to pity, released them both; and by this deed Wang Lin's name became renowned throughout his hometown.
#: '''Original:''' 汝南有王琳巨尉者，年十余岁丧父母。因遭大乱，百姓奔逃，惟琳兄弟独守冢庐，号泣不绝。弟季，出遇赤眉，将为所哺，琳自缚，请先季死，贼矜而放遣，由是显名乡邑。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Wei Tan Spares His Fellow Captives, ''Hou Han Shu''
#: '''English''': Wei Tan of Langye, styled Shaoxian, was likewise seized by starved bandits. Several dozen captives were bound and awaited their turn to be boiled. 
#: The bandits, seeing that Tan appeared honest and trustworthy, set him apart to tend the cooking fire, though they bound him again each evening. Among the bandits was one Yi Changgong, who took especial pity on Tan; he secretly loosened Tan's bonds and said: &quot;Ye are all destined to be eaten; flee hence at once.&quot; 
#: Tan replied: &quot;I have tended the fire for ye, there I always had some leavings for myself; the others have been fed only on grass and weeds; better to eat (''relatively well-fed'') me instead.&quot; Changgong, moved by his righteousness, persuaded the others to release all the captives, and all were spared.
#: '''Original:''' 琅邪魏谭少闲者，时亦为饥寇所获，等辈数十人皆束缚，以次当亨（通“烹”）。贼见谭似谨厚，独令主爨，暮辄执缚。贼有夷长公，特哀念谭，密解其缚，语曰：&quot;汝曹皆应就食，急从此去。&quot;对曰：&quot;谭为诸君爨，恒得遗余，余人皆菇草莱，不如食我。&quot;长公义之，相晓赦遣，并得俱免。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Er Meng and Che Cheng Offer Themselves for Each Other, ''Hou Han Shu''
#: '''English''': Er Meng Ziming of Qi and Che Cheng Ziwei of Liangjun, brothers, were seized together by the Red Eyebrows and were about to be eaten. Meng and Cheng knelt and each begged to die in the other's stead. The bandits, moved to pity, released them both.
#: '''Original:''' 齐国兒萌子明、梁郡车成子威二人，兄弟并见执于赤眉，将食之，萌、成叩头，乞以身代，贼亦哀而两释焉。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)
# Chunyu Gong Offers Himself for His Brother, ''Hou Han Shu''
#: '''English''': Chunyu Gong, styled Mengsun, was a man of Chunyu in Beihai. […] At the end of Wang Mang's reign, when famine and war arose, his elder brother Chong was seized by bandits who meant to boil and eat him. Gong begged to take his brother's place; both were released.
#: '''Original:''' 淳于恭字孟孙，北海淳于人也。[…] 王莽末，岁饥兵起，恭兄崇将为盗所亨，恭请代，得俱免。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu, Zhao, Chunyu, Jiang, Liu, Zhou and Zhao, Vol. 39&quot; (《後漢書·卷三十九·劉趙淳于江劉周趙列傳第二十九》)

== Three Kingdoms period ==
According to population statics at the time, the population of the Three Kingdoms period was only one-seventh of that during the reign of Emperor Huan of the Eastern Han Dynasty.&lt;ref&gt;秦晖，《中国历史上，何来如此深仇大恨》&lt;/ref&gt; This was the largest population decrease in Chinese history, evidenced by Cao Cao's poem; &quot;Pale bones exposed in wild fields, no crowing of roosters heard throughout thousands of li&quot; (白骨露于野，千里无鸡鸣).
# 194 CE: Famine During the Puyang Campaign, ''Sanguozhi''
#: '''English''': That year, one hu of grain fetched over fifty thousand coins; people ate each other. Newly recruited troops were thereupon disbanded.
#: '''Original:''' 是岁谷一斛五十余万钱，人相食，乃罢吏兵新募者。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Annals of Emperor Wu, Vol. 1&quot; (《三國志·卷一·魏書一·武帝紀》)
## 194 CE: Famine During the Puyang Campaign, ''Sanguozhi(2)''
##: '''English''': Cao Cao led his forces back and gave battle to Lü Bu at Puyang; his army fared ill and the two sides held their positions for over a hundred days. That year brought drought, locusts and scarcity of grain; the people ate each other. Lü Bu withdrew eastward to encamp at Shanyang.
##: '''Original:''' 太祖引军还，与布战于濮阳，太祖军不利，相持百余日。是时岁旱、虫蝗、少谷，百姓相食，布东屯山阳。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Lü Bu, Vol. 7&quot; (《三國志·卷七·魏書七·呂布臧洪傳》)
## 194 CE: Famine During the Puyang Campaign, ''Sanguozhi(3)''
##: '''English''': Cao Cao and Lü Bu held their positions at Puyang; Sima Lang thereupon led his household back to Wen. That year brought Great Famine; people ate each other. Lang gathered and succoured his kinsmen, tutored his younger brothers, and did not abandon his studies in that age of decline.
##: '''Original:''' 时岁大饥，人相食，朗收恤宗族，教训诸弟，不为衰世解业。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Sima Lang, Vol. 15&quot; (《三國志·卷十五·魏書十五·劉司馬梁張溫賈傳》)
## 194 CE: Famine During the Puyang Campaign, ''Hou Han Shu''
##: '''English''': Cao Cao heard of this and led his forces to attack Lü Bu; they fought repeatedly and held their positions for over a hundred days. That year brought drought, locusts and scarcity of grain; the people ate each other. Lü Bu withdrew to encamp at Shanyang.
##: '''Original:''' 曹操闻而引军击布，累战，相持百余日。是时，旱、蝗，少谷，百姓相食，布移屯山阳。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yan, Yuan Shu and Lü Bu, Vol. 75&quot; (《後漢書·卷七十五·劉焉袁術呂布列傳第六十五》)
# 194 CE: Cheng Yu's Human Jerky, Pei Songzhi's Commentary
#: '''English''': In the beginning, Cao Cao's forces lacked provisions. 
#: Cheng Yu seized supplies from his home county to provide three days' rations, mixed in no small part with dried human flesh. By this reason, he lost the favour of the ''(heavenly)'' court, and therefore never attained the rank of the Excellencies.
#: '''Original:''' 初，太祖乏食；昱略其本县，供三日粮，颇杂以人脯。由是失朝望，故位不至公。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Weijin Shiyu'', &quot;Biography of Cheng Yu, Vol. 14&quot; (裴松之《三國志注·卷十四·魏書十四·程昱傳》引《魏晉世語》)
# 195 CE: Great Famine at Chengshi, ''Sanguozhi''
#: '''English''': Cao Cao's forces were stationed at Chengshi. Great Famine; people ate each other.
#: '''Original:''' 太祖军乘氏，大饥，人相食。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biography of Xun Yu, Vol. 10&quot; (《三國志·卷十·魏書十·荀彧荀攸賈詡傳》)
# 195 CE: The Siege of Dongjun, ''Hou Han Shu''
#: '''English''': [...] At first they still dug for rats and boiled sinew and hide; afterwards there was nothing left to eat.
#: The chief clerk reported that there were three dou of rice in the inner kitchen and requested it be made into gruel. Zang Hong said: &quot;How could I alone enjoy this?&quot; He had it made into thin porridge and distributed among all the troops.
#: He also slew all his beloved concubine to feed his officers and men. The officers and men all wept; none could raise their eyes to look upon him. Seventy or eighty men and women died lying upon one another; not one deserted or betrayed him.
#: '''Original:''' （东郡）初尚掘鼠，煮筋角，后无所复食，主簿启内厨米三斗，请稍为饘粥，洪曰：&quot;何能独甘此邪？&quot;使为薄糜，遍班士众。又杀其爱妾，以食兵将。兵将咸流涕，无能仰视。男女七八十人相枕而死，莫有离叛。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biography of Zang Hong, Vol. 58&quot; (《後漢書·卷五十八·虞詡等列傳》)
# 195 CE: The Siege of Dongjun, ''Zizhi Tongjian''
#: '''English''': At first they still dug for rats and boiled sinew and hide; afterwards there was nothing left to eat. 
#: The chief clerk reported only three sheng of rice in the inner kitchen and requested it be made into gruel. Zang Hong sighed: &quot;How could I alone enjoy this!&quot; He had it made into thin porridge and distributed among all the troops; he also slew his beloved concubine to feed his officers and men. 
#: The officers and men all wept; none could raise their eyes to look upon him. Seven or eight thousand men and women died lying upon one another; not one deserted or betrayed him.
#: '''Original:''' 初尚掘鼠煮筋角，后无可复食者。主簿启内厨米三升，请稍以为饘粥，臧洪叹曰：&quot;何能独甘此邪！&quot;使作薄糜，遍班士众，又杀其爱妾以食将士。将士咸流涕，无能仰视者。男女七八千人，相枕而死，莫有离叛者。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Hou Han Shu''
#: '''English''': [...] One hu of grain fetched fifty thousand coins; beans and wheat twenty thousand. People ate each other; white bones lay heaped in piles, and the stench of rot filled the roads. [...] After Li Jue and Guo Si turned upon each other and the Son of Heaven departed eastward, Chang'an stood empty for over forty days. The strong scattered; the weak ate each other. Within two or three years, not a human trace remained in Guanzhong.
#: '''Original:''' 自（李）傕、（郭）汜相攻，天子东归后，是时，谷一斛五十万，豆、麦二十万，人相食啖，白骨委积，臭秽满路。……长安城空四十余日，强者四散，蠃者相食，二三年间，关中无复人迹。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biography of Dong Zhuo, Vol. 72&quot; (《後漢書·卷七十二·董卓列傳第六十二》)
## 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Sanguozhi''
##: '''English''': At that time the three adjuncts still held several hundred thousand households. Li Jue and his confederates unleashed their troops to plunder, attacking and pillaging cities and towns. The people suffered Great Famine; within two years they had eaten each other to the last.
##: '''Original:''' 时三辅民尚数十万户，傕等放兵劫略，攻剽城邑，人民饥困，二年间相啖食略尽。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of Dong, the Two Yuans and Liu, Vol. 6&quot; (《三國志·卷六·魏書六·董二袁劉傳》)
## 195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Jin Shu''
##: '''English''': [...] One hu of grain fetched fifty thousand coins; beans and wheat twenty thousand. People ate each other; white bones lay heaped in great mounds, the rotting remains befouling the roads. [...] Chang'an stood entirely empty; all scattered to the four winds. Within two or three years, not a traveller remained in Guanzhong. [...] Since Dong Zhuo's rebellion, the people had been scattered and adrift; grain reached over fifty thousand coins per shi, and many ate each other.
##: '''Original:''' 是时谷一斛五十万，豆麦二十万，人相食啖，白骨盈积，残骸余肉，臭秽道路。……长安城中尽空，并皆四散，二三年间，关中无复行人。……汉自董卓之乱，百姓流离，谷石至五十余万，人多相食。
##: '''Source:''' ''[[:w:Jin Shu|Jin Shu]]'', &quot;Treatise on Food and Commerce, Vol. 26&quot; (《晉書·卷二十六·志第十六·食貨》)
##195–197 CE: The Chaos of Li Jue and Guo Si in Chang'an, ''Zizhi Tongjian''
##: '''English''': When Dong Zhuo first died, the three adjuncts still held several hundred thousand households. Li Jue and his confederates unleashed their troops to plunder; compounded by Great Famine, within two years the people had eaten each other nearly to the last. 
##: [...] At that time Chang'an stood empty for over forty days; the strong scattered, the weak ate each other, and within two or three years not a human trace remained in Guanzhong.
##: '''Original:''' 董卓初死，三辅民尚数十万户，李傕等放兵劫略，加以饥馑，二年间，民相食略尽。……是时，长安城空四十馀日，强者四散，羸者相食，二三年间，关中无复人迹。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 61 (《資治通鑑·卷六十一》)
# 195–197 CE: Wang Zhong the Cannibal, Pei Songzhi's Commentary
#: '''English''': Wang Zhong was a man of Fufeng who in his youth served as a village headman. When the three adjuncts fell into turmoil, Zhong, starving and desperate, ate human flesh, and followed a band of men southward toward Wuguan. [...]
#: The Master of the Wuguan Office, knowing that Zhong had once eaten human flesh, took him along on an imperial outing and had entertainers fasten a skull from a grave to Zhong's saddle, to the great amusement of all.
#: '''Original:''' 王忠，扶风人。少为亭长。三辅乱，忠饥乏噉人，随辈南向武关。……五官将知忠尝噉人，因从驾出行，令俳取冢间骷髅系著忠马鞍，以为欢笑。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Weilüe'', &quot;Annals of Emperor Wu, Vol. 1&quot; (裴松之《三國志注·魏書·武帝紀》引《魏略》)
# 196 CE: Liu Bei's Army Starves at Haixi, Zizhi Tongjian
#: '''English''': Liu Bei gathered his remaining forces and moved east to Guangling, gave battle to Yuan Shu, and was again defeated; he encamped at Haixi. Beset by hunger and hardship, his officers and men ate each other.
#: '''Original:''' 备收馀兵东取广陵，与袁术战，又败，屯于海西。饥饿困踧，吏士相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 62 (《資治通鑑·卷六十二》)
# 196 CE: Liu Bei's Army Starves at Haixi, Pei Songzhi's Commentary
#: '''English''': Liu Bei's army being at Guangling, hunger and hardship upon them; officers and men, high and low, ate each other.
#: '''Original:''' 備軍在廣陵，飢餓困踧，吏士大小自相啖食。
#: '''Source:''' Pei Songzhi's ''[[:w:Annotations to the Records of the Three Kingdoms|Sanguozhi Annotations]]'', citing the lost ''Yingxiong Ji'', &quot;Biography of the Progenitor Ruler, Vol. 32&quot; (裴松之《三國志注·卷三十二·蜀書·先主傳》引《英雄記》)
# 196 CE: Famine Under Gongsun Zan's Rule, ''Hou Han Shu''
#: '''English''': [...] That year brought drought and locusts; grain grew dear and people ate each other. Gongsun Zan, relying on his own abilities, showed no concern for the people.
#: '''Original:''' 是时，旱、蝗，谷贵，民相食。瓒恃其才力，不恤百姓。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yu, Gongsun Zan and Tao Qian, Vol. 73&quot; (《後漢書·卷七十三·劉虞公孫瓚陶謙列傳第六十三》)
# 197 CE: Famine Along the Yangtze and Huai, ''Hou Han Shu(1)''
#: '''English''': That year brought famine; along the Yangtze and Huai rivers, people ate each other.
#: '''Original:''' 是岁饥，江淮间民相食。
#: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Annals of Emperor Xian, Vol. 9&quot; (《後漢書·卷九·孝獻帝紀第九》)
## 197 CE: Famine Along the Yangtze and Huai, ''Hou Han Shu(2)''
##: '''English''': Yuan Shu's forces were weakened, his great generals dead, and his followers estranged and in revolt. Compounded by drought and failed harvests, his officers and people froze and starved; along the Yangtze and Huai, people had eaten each other nearly to the last.
##: '''Original:''' 术兵弱，大将死，众情离叛，加天旱岁荒，士民冻馁，江、淮间相食殆尽。
##: '''Source:''' ''[[:w:Book of the Later Han|Hou Han Shu]]'', &quot;Biographies of Liu Yan, Yuan Shu and Lü Bu, Vol. 75&quot; (《後漢書·卷七十五·劉焉袁術呂布列傳第六十五》)
## 197 CE: Famine Along the Yangtze and Huai, ''Sanguozhi''
##: '''English''': Yuan Shu's extravagance grew ever more excessive; his rear palace of several hundred consorts all wore fine silks, with surplus of grain and meat, whilst his officers and men froze and starved. Along the Yangtze and Huai the land was emptied; people ate each other.
##: '''Original:''' 荒侈滋甚，后宫数百皆服绮縠，余粱肉，而士卒冻馁，江淮间空尽，人民相食。
##: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of Dong, the Two Yuans and Liu, Vol. 6&quot; (《三國志·卷六·魏書六·董二袁劉傳》)
## 197 CE: Famine Along the Yangtze and Huai, ''Zizhi Tongjian''
##: '''English''': Since the Zhongping era, the realm had fallen into turmoil; the people abandoned farming, armies rose on all sides, and provisions were ever wanting. When hungry, the troops plundered; when fed, they abandoned their surplus. Those who collapsed and scattered, undone by no enemy but themselves, were beyond counting. Yuan Shao in Hebei had his men subsist on mulberries; Yuan Shu along the Yangtze and Huai drew sustenance from cattail and river snails. The people ate each other, and the commanderies were left desolate.
##: '''Original:''' 中平以来，天下乱离，民弃农业，诸军并起，率乏粮谷，无终岁之计，饥则寇略，饱则弃馀，瓦解流离，无敌自破者，不可胜数。袁绍在河北，军人仰食桑椹。袁术在江淮，取给蒲蠃，民多相食，州里萧条。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 62 (《資治通鑑·卷六十二》)
# 238 CE: Siege of Xiangping. ''Sanguozhi''
#: '''English''': Gongsun Yuan was in dire stuation. His provisions exhausted, people ate each other, and the dead were very many.
#: '''Original:''' 渊窘急。粮尽，人相食，死者甚多。
#: '''Source:''' ''[[:w:Sanguozhi|Sanguozhi]]'', &quot;Biographies of the Two Gongsuns, Tao and Four Zhangs, Vol. 8&quot; (《三國志·卷八·魏書八·二公孫陶四張傳》)
## 238 CE: Siege of Xiangping, ''Zizhi Tongjian''
##: '''English''': Gongsun Yuan was in dire situation; provisions in Xiangping were exhausted, people ate each other, and the dead were very many.
##: '''Original:''' 公孙渊窘急，粮尽，人相食，死者甚多。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 74 (《資治通鑑·卷七十四》)

==West Jin==

# 304 CE: The Famine of Chang'an and the Sack of Luoyang, ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': Shen Ju raised arms against Chang'an, yet was routed by (Sima) Yong. Zhang Fang greatly plundered Luo, then withdrew unto Chang'an. Thereupon the armies fell into dire want, and men did eat one another.
#: '''Original:''' 沈举举兵攻长安，为（司马）颙所败。张方大掠洛中，还长安。于是军中大馁，人相食。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals of Emperor Hui&quot; (《晋书·卷四·帝纪第四·惠帝》)
# 304 CE: The Plunder of Luoyang, in ''[[w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': (Zhang Fang) did seize from Luo above ten thousand bondsmen and bondswomen, both of state and private households, and marched them westward. The army, lacking victuals, did slay men and mingle their flesh with that of oxen and horses for sustenance.
#: '''Original:''' （张方）掠洛中官私奴婢万馀人而西。军中乏食，杀人杂牛马肉食之。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 85 (《资治通鉴》卷85)
# 306 CE: The Tyranny of Pan Tao and Bi Miao, in ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': (Pan) Tao and (Bi) Miao and their like seized (Sima) Yue and force him beyond the passes, falsely establishing a mobile administration, compelling the removal of ministers, issuing decrees by their own will, loosing soldiers to plunder and ravage, consuming the flesh of the common people, with corpses choking the roads and bleached bones filling the wilderness. Thus did the provincial lords betrayed their obligation, the cities and towns fall desolate, and the folk of Huai and Yu were casted into utter misery.
#: '''Original:''' （潘）滔、（毕）邈等劫（司马）越出关，矫立行台，逼徙公卿，擅为诏令，纵兵寇抄，茹食居人，交尸塞路，暴骨盈野。遂令方镇失职，城邑萧条，淮豫之萌，陷离涂炭。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biography of Zhou Jun et al.&quot; (《晋书·卷六十一·列传第三十一·周浚等》)
# 311 CE, eign of [[:w:Emperor Huai of Jin|Emperor Huai of Jin]]: The Rout at Ningping and the Death of Sima Yue, in ''[[w:Book of Jin|Book of Jin]]''
#: '''English''': In the fifth year of Yongjia (the third month), (Sima) Yue did perish at Xiang. In the fourth month, Shi Le gave pursuit unto Ningping in Ku County; General Qian Duan sallied forth to resist him and fell in battle, the army breaking asunder. Thereupon Shi Le encircled the host of several hundred thousand with cavalry and loosed arrows upon them; the slain were heaped as mountains. Of princes, nobles, officers, and commoners, above a hundred thousand perished. Wang Mi's brother Zhang did burn the remnant and devour them. 
#: The people laid blame upon (Sima) Yue, and Emperor Huai issued a decree degrading Yue to the rank of a county king.
#: '''Original:''' 永嘉五年（三月），（司马越）薨于项。……（四月，）石勒追及于苦县宁平城，将军钱端出兵距勒，战死，军溃。……于是数十万众，（石）勒以骑围而射之，相践如山。王公士庶死者十余万。王弥弟璋焚其余众，并食之。天下归罪于（司马）越。（晋怀）帝发诏贬越为县王。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin (晉書)]]'', &quot;Biography of King Liang of Runan et al.&quot; (《晋书·卷五十九·列传第二十九·汝南王亮等》)
# 311 CE, Reign of [[:w:Emperor Huai of Jin|Emperor Huai of Jin]]: The Famine in the Passes, in ''[[w:Book of Jin|Book of Jin]](1)''
#: '''English''': At that time, famine ravaged the lands within the passes; the common folk consumed ate each other. Pestilence spreaded upon them, and bandits roamed openly, beyond the power of (Sima) Mo to suppress.
#: '''Original:''' 時關中饑荒，百姓相啖；加以疾疫，盜賊公行，（司马）模力不能制。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin (晉書)]]'', &quot;Biographies of the Imperial Clan&quot; (《晋书·卷三十七·列传第七·宗室》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](2)''
##: '''English''': Grand General Xun Xi memorialized to relocate the capital to Cangyuan; the Emperor was minded to comply, yet the great ministers, fearing (Pan) Tao, dared not carry out the edict, and the palace eunuchs, coveting their riches, were loath to depart. Famine grew great; people ate each other, and eight or nine in ten officials fled.
##: '''Original:''' 大将军苟晞表迁都仓垣,帝将从之,诸大臣畏滔,不敢奉诏,且宫中及黄门恋资财,不欲出。至是饥甚,人相食,百官流亡者十八九。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals, Emperor Huai &amp; Emperor Min&quot; (《晋书·卷五·帝纪第五·孝怀帝 孝愍帝》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](3)''
##: '''English''': By the Yongjia era, calamity and disorder had worsened greatly. East of Yongzhou, multitudes suffered hunger; they sold one another into bondage, and the wandering multitudes were beyond count. Six provinces — You, Bing, Si, Ji, Qin, and Yong — were struck by great locusts, devouring all grass, trees, and the fur of cattle and horses. Great pestilence followed, joined by famine. People were slain by brigands; corpses filled the rivers, and white bones covered the fields. As Liu Yao's forces pressed close, the court deliberated removing the capital to Cangyuan. People ate each other; famine and plague came together, and eight or nine in ten officials had fled.
##: '''Original:''' 至于永嘉,丧乱弥甚。雍州以东,人多饥乏,更相鬻卖,奔迸流移,不可胜数。幽、并、司、冀、秦、雍六州大蝗,草木及牛马毛皆尽。又大疾疫,兼以饥馑。百姓又为寇贼所杀,流尸满河,白骨蔽野。刘曜之逼,朝廷议欲迁都仓垣。人多相食,饥疫总至,百官流亡者十八九。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Treatise on Food and Commerce&quot; (《晋书·卷二十六·志第十六·食货》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](3)''
##: '''English''': Emperor Huai being besieged by Liu Yao, the imperial armies suffered repeated defeat, the treasury was exhausted, and the hundred officials were greatly famished; smoke of cooking fires was seen in no house. The starving fed upon one another. In the west, where Emperor Min resided, hunger was exceeding great; a peck of grain cost two taels of gold, and more than half the people perished.
##: '''Original:''' 怀帝为刘曜所围,王师累败,府帑既竭,百官饥甚,比屋不见火烟,饥人自相啖食。愍皇西宅,馁馑弘多,斗米二金,死者太半。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Treatise on Food and Commerce&quot; (《晋书·卷二十六·志第十六·食货》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](4)''
##: '''English''': When Luoyang fell into chaos, with thieves running rampant, people ate each other out of hunger. (Zhi) Yu, being ever poor and frugal, perished at last of starvation.
##: '''Original:''' 及洛京荒乱,盗窃纵横,人饥相食。虞素清贫,遂以馁卒。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Huangfu Mi et al.&quot; (《晋书·卷五十一·列传第二十一·皇甫谧等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](5)''
##: '''English''': (Wang) Mi, together with (Liu) Yao, attacked Xiangcheng and pressed upon the capital. The capital suffered a Great Famine; people ate each other, the common folk fled, and the dukes and ministers escaped to Heyin.
##: '''Original:''' 王弥后与曜寇襄城,遂逼京师。时京邑大饥,人相食,百姓流亡,公卿奔河阴。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Wang Mi et al.&quot; (《晋书·卷一百·列传第七十·王弥等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]](6)''
##: '''English''': Wang Mi and Liu Yao arrived and joined (Huyan) Yan in besieging Luoyang. Within the city, famine was dire; people ate each other, the hundred officials scattered, and none held firm. The Xuanyang Gate fell; Mi and Yan entered the Southern Palace, ascended the Taiji Front Hall, and loosed their soldiers in great plunder, seizing all palace women and treasures. Yao thereupon slew all the princes, nobles, and officers below, in which numbered more than thirty thousand in all, and thereupon raised a great mound of their skulls north of the Luo River.
##: '''Original:''' 王弥、刘曜至,复与晏会围洛阳。时城内饥甚,人皆相食,百官分散,莫有固志。宣阳门陷,弥、晏入于南宫,升太极前殿,纵兵大掠,悉收宫人、珍宝。曜于是害诸王公及百官已下三万余人,于洛水北筑为京观。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)
## 311 CE, Reign of Emperor Huai of Jin: Great Famine and Cannibalism During the Fall of Luoyang, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': Ere long, Luoyang fell to famine and distress; people ate each other, and eight or nine in ten officials had fled.
##: '''Original:''' 既而洛阳饥困,人相食,百官流亡者什八九。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 87 (《资治通鉴》卷87)
# 311 CE, Reign of Emperor Huai of Jin (永嘉五年): Great Famine and Cannibalism After the Fall of Luoyang, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': When Luoyang fell, Grand Commandant Xun Fan fled to Yangcheng, and General of the Guard Hua Hui fled to Chenggao. A Great Famine prevailed; the bandit chief Hou Du and his ilk seized men for food, and many of Fan's and Hui's followers were thus devoured.
#: '''Original:''' 及洛阳不守，太尉荀藩奔阳城，卫将军华荟奔成皋。时大饥，贼帅侯都等每略人而食之，藩、荟部曲多为所啖。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Shao Xu et al.&quot; (《晋书·卷六十三·列传第三十三·邵续等》)
# 312 CE: Cannibalism Among Han Zhao Troops, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': The Han Zhao generals Zhao Gu and Wang Sang, fearing absorption by Shi Le, sought to lead their forces back to Pingyang. Provisions within the army ran short, and soldiers ate each other.
#: '''Original:''' 汉安北将军赵固、平北将军王桑恐为石勒所并，欲引兵归平阳。军中乏粮，士卒相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 88 (《资治通鉴》卷88)
# 312 CE: Cannibalism in Shi Le's Army at Gepei, in ''[[:w:Book of Jin|Book of Jin]]'' and ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': (Shi) Le, at Gepei, built dwellings, encouraged farming, and constructed boats, intending to attack Jiankang. Yet wherever he marched, the people had fortified their walls and cleared the fields; nothing could be plundered, and great famine fell upon the army, so that soldiers ate each other.
#: '''Original:''' 勒于葛陂缮室宇，课农造舟，将寇建邺。……勒所过路次，皆坚壁清野，采掠无所获，军中大饥，士众相食。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Shi Le I&quot; (《晋书·卷一百四·载记第四·石勒上》)
# 312 CE: Cannibalism in Shi Le's Army at Gepei, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
#: '''English''': As Shi Le marched north from Gepei, all along his path the people had fortified and cleared the fields; nothing could be seized. Famine within the army grew dire, and soldiers ate each other.
#: '''Original:''' 石勒自葛陂北行，所过皆坚壁清野，虏掠无所获，军中饥甚，士卒相食。
#: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 88 (《资治通鉴》卷88)
# 314 CE: Monstrous Birth and Cannibalism in Guangyi, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': The wife of Yang Chong of Guangyi bore a child with two heads; her brother stole and ate it, and died within three days.
#: '''Original:''' 光义人羊充妻产子二头，其兄窃而食之，三日而死。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)
# 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Book of Jin|Book of Jin]](1)''
#: '''English''': In the tenth month of winter, the capital Chang'an suffered dire famine; a peck of grain cost two taels of gold, people ate each other, and more than half perished.
#: '''Original:''' 冬十月，京师饥甚，米斗金二两，人相食，死者太半。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Annals, Emperor Huai &amp; Emperor Min&quot; (《晋书·卷五·帝纪第五·孝怀帝 孝愍帝》)
## 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Book of Jin|Book of Jin]](2)''
##: '''English''': When Liu Yao again besieged the capital, (Suo) Chen and Qu Yun held fast to the inner city of Chang'an. Within, famine was dire; people ate each other, and the dead, fugitives, and deserters were beyond restraint; only the thousand loyal troops from Liangzhou stood firm unto death.
##: '''Original:''' 后刘曜又率众围京城、綝与麹允固守长安小城。……城中饥窘，人相食，死亡逃奔不可制，唯凉州义众千人守死不移。
##: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', &quot;Biographies, Xie Xi et al.&quot; (《晋书·卷六十·列传第三十·解系等》)
## 316 CE, Reign of Emperor Min of Jin: Great Famine and Cannibalism at Chang'an, in ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]''
##: '''English''': In the eighth month, the Han Zhao Grand Marshal (Liu) Yao pressed upon Chang'an. Yao stormed the outer city; Qu Yun and Suo Chen withdrew to defend the inner city. All communication within and without was severed; famine within grew dire. A peck of grain cost two taels of gold, people ate each other, and more than half had perished; deserters and fugitives could not be restrained. Only the thousand loyal troops from Liangzhou stood firm. In the imperial granary there remained but several dozen cakes of leaven; Qu Yun ground them into gruel to feed the Emperor, yet ere long even these were exhausted.
##: '''Original:''' 八月，汉大司马曜逼长安。……曜攻陷長安外城，麴允、索綝退保小城以自固。內外斷絕，城中饑甚。斗米值金二兩，人相食，死者大半，亡逃不可制。唯涼州義眾千人守死不移。太倉有麴數十餅，麴允屑之為粥以供帝，既而亦盡。
##: '''Source:''' ''[[:w:Zizhi Tongjian|Zizhi Tongjian]]'', Vol. 89 (《资治通鉴》卷89)
# 316 CE: Great Famine and Cannibalism in Beidi, in ''[[:w:Book of Jin|Book of Jin]]''
#: '''English''': Famine in Beidi was dire; people ate each other. Qiang Qiou's army transported grain to supply Qu Chang, but was defeated by Liu Ya.
#: '''Original:''' 北地饥甚，人相食啖，羌酋大军须运粮以给麹昌，刘雅击败之。
#: '''Source:''' ''[[:w:Book of Jin|Book of Jin]]'', Vol. 102 &quot;Chronicles, Liu Cong et al.&quot; (《晋书·卷一百二·载记第二·刘聪等》)

==East Jin==
# 319 CE: Slicing and Eating of Du Zeng's Flesh, ''Book of Jin''
#: '''English''': Du Zeng's forces collapsed; his generals Ma Jun and Su Wen captured him and surrendered to Zhou Fang. Zhou Fang wished to bring him alive to Wuchang, but Zhu Gui's son Zhu Chang and Zhao You's son Zhao Yin both begged for Du Zeng to avenge their fathers' grievances. Du Zeng was thereupon beheaded; Chang and Yin sliced his flesh and ate it.
#: '''Original:''' 曾众溃，其将马俊、苏温等执曾诣访降。访欲生致武昌，而朱轨息昌、赵诱息胤皆乞曾以复冤，于是斩杜曾，而昌、胤脔其肉而啖之。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 100, &quot;Biographies, Vol. 70: Wang Mi et al.&quot; (《晋书·卷一百·列传第七十·王弥等》)
# c. 321 CE: Xu Kan Fed to His Own Kin After Execution, ''Book of Jin''
#: '''English''': Shi Jilong attacked and captured Xu Kan, sending him to Xiangguo. Shi Le had him bagged and hurled to his death from the hundred-foot tower, then ordered the wives and children of Bu Du and others to disembowel and eat him; three thousand of Xu Kan's surrendered troops were buried alive.
#: '''Original:''' 石季龙攻陷徐龛，送之襄国，勒囊盛于百尺楼自上扑杀之，令步都等妻子刳而食之，坑龛降卒三千。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 105, &quot;Chronicles, Vol. 5: Shi Le, Part II et al.&quot; (《晋书·卷一百五·载记第五·石勒下等》)
# c. 337 CE: Shi Sui Slays Palace Women and Nuns, ''Book of Jin(1)''
#: '''English''': After Shi Sui assumed full governance, he abandoned himself to wine and lust, acting with arrogant depravity. He would roam the fields with music playing as he entered, or venture by night into the homes of court officials to violate their wives and concubines. 
#: Of the palace women whom he had adorned and found comely, he would behead them, wash away the blood, place their heads upon platters, and pass them round for viewing. He also brought in comely Buddhist nuns, defiled them, then slew them; their flesh was boiled together with beef and mutton and eaten, and portions were also distributed to his attendants, who were interested in the flavor.
#: '''Original:''' 邃自总百揆之后，荒酒淫色，骄恣无道，或盘游于田，悬管而入，或夜出于宫臣家，淫其妻妾。妆饰宫人美淑者，斩首洗血，置于盘上，传共视之。又内诸比丘尼有姿色者，与其交亵而杀之，合牛羊肉煮而食之，亦赐左右，欲以识其味也。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 106, &quot;Chronicles, Vol. 6: Shi Jilong, Part I&quot; (《晋书·卷一百六·载记第六·石季龙上》)
## c. 337 CE: Shi Sui Slays and Cooks Palace Women and Nuns, ''Zizhi Tongjian''
##: '''English''': Shi Sui, Crown Prince of Later Zhao, was arrogant, lustful, and cruel; he delighted in adorning comely consorts, beheading them, washing away the blood, placing their heads upon platters, and passing them amongst his guests for viewing. He further cooked their flesh and shared it for eating.
##: '''Original:''' 邃骄淫残忍，好妆饰美姬，斩其首，洗血置盘上，与宾客传观之，又烹其肉共食之。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 95 (《资治通鉴》卷95)
# 351 CE: Great Famine in Si and Ji Provinces, ''Book of Jin(1)''
#: '''English''': Bandits and rebels arose like swarms; a Great Famine struck Si and Ji Provinces; people ate each other. 
#: From the final years of Shi Jilong, Ran Min had dispersed all the granaries and treasuries to cultivate personal loyalty. Warfare with the Qiang and Hu raged without cease, with battles every month. 
#: The transplanted households of Qing, Yong, You, and Jing Provinces, together with the Di, Qiang, Hu, and Man peoples, numbering several hundred myriads, returned to their native lands; their routes met in one point, where all of they slaughtered and plundered one another. With famine and pestilence, only two or three in ten reached their destinations. Throughout the realm there was great disorder, and none remained to till the fields.
#: '''Original:''' 贼盗蜂起，司、冀大饥，人相食。自季龙末年而闵尽散仓库以树私恩。与羌胡相攻，无月不战。青、雍、幽、荆州徙户及诸氐、羌、胡、蛮数百余万，各还本土，道路交错，互相杀掠，且饥疫死亡，其能达者十有二三。诸夏纷乱，无复农者。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 107, &quot;Chronicles, Vol. 7: Shi Jilong, Part II&quot; (《晋书·卷一百七·载记第七·石季龙下》)
## 351 CE: Great Famine in Si and Ji Provinces, ''Zizhi Tongjian''
##: '''English''': The several hundred myriad transplanted peoples of Qing, Yong, You, and Jing Provinces — along with the Di, Qiang, Hu, and Man — whom Later Zhao had relocated, found the laws of Zhao no longer enforced and each returned to their native lands.
##: Their routes met in one point, where all of they slaughtered and plundered one another; only two or three in ten reached their destinations. The Central Plains fell into great disorder. Famine and pestilence followed; people ate each other, and none remained to till the fields.
##: '''Original:''' 后赵所徙青、雍、幽、荆四州人民及氐、羌、胡蛮数百万口，以赵法禁不行，各还本土；道路交错，互相杀掠，其能达者什有二、三。中原大乱。因以饥疫，人相食，无复耕者。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 99 (《资治通鉴》卷99)
# 352 CE: Famine in Ye, ''Book of Jin''
#: '''English''': Famine struck Ye; people ate each other. The palace women from the time of Shi Jilong were nearly all consumed.
#: '''Original:''' 邺中饥，人相食，季龙时宫人被食略尽。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 107, &quot;Chronicles, Vol. 7: Shi Jilong, Part II&quot; (《晋书·卷一百七·载记第七·石季龙下》)
## 352 CE: Famine in Ye'', Zizhi Tongjian''
##: '''English''': A Great Famine struck Ye; people ate each other. The palace women from the time of the former Zhao were nearly all consumed.
##: '''Original:''' 邺中大饥，人相食，故赵时宫人被食略尽。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 99 (《资治通鉴》卷99)
# 356 CE: Siege of Duan Kan's City, ''Zizhi Tongjian''
#: '''English''': Duan Kan defended the Yin city under siege; the roads for gathering firewood were cut off, and people ate each other within the city.
#: '''Original:''' 段龛婴城自守，樵采路绝，城中人相食。
#: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 100 (《资治通鉴·卷一百》)
# 385 CE: Great Famine at Chang'an, ''Book of Jin''
#: '''English''': At this time there was a Great Famine in Chang'an; people ate each other, and the generals, upon returning home, spat out flesh to feed their wives and children.
#: '''Original:''' 时长安大饥，人相食，诸将归而吐肉以饴妻子。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
## 385 CE: Great Famine at Chang'an, ''Wei Shu''
##: '''English''': Great Famine in Chang'an; people ate each other. Yao Chang rebelled at Beidi and allied with [Murong] Chong, jointly attacking Chang'an.
##: '''Original:''' 长安大饥，人民相食。姚苌叛于北地，与冲连和，合攻长安。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 95 &quot;Biographies 83, Liu Cong of the Xiongnu et al.&quot; (《魏书·卷九十五·列传第八十三·匈奴刘聪等》)
## 385 CE: Great Famine at Chang'an, ''Zizhi Tongjian''
##: '''English''': In the first month, [Former] Qin's [Fu] Jian held a banquet for his ministers. Chang'an was then stricken by famine; people ate each other, and the generals, upon returning home, spat out flesh to feed their wives and children.
##: '''Original:''' 正月，(前)秦(苻)堅朝饗群臣，時長安飢，人相食，諸將歸，吐肉以飼妻子。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 106 (《资治通鉴·卷一百零六》)
# 385 CE: Murong Chong's Forces Eat the Slain, ''Book of Jin''
#: '''English''': [Murong] Chong further dispatched his Secretariat Director Gao Gai to lead troops in a night assault on Chang'an, breaching the southern gate and entering the southern city. General of the Left Dou Chong and General of the Front Guards Li Bian and others repelled them, beheading 1,800 men, and divided the corpses for consumption.
#: '''Original:''' （慕容）冲又遣其尚书令高盖率众夜袭长安，攻陷南门，入于南城。左将军窦冲、前禁将军李辩等击败之，斩首千八百级，分其尸而食之。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
# 385 CE: Famine in You and Ji Prefectures, ''Book of Jin''
#: '''English''': Murong Gui's troops suffered greatly from hunger and many fled to Zhongshan; the people of You and Ji prefectures ate each other. Earlier, a popular rhyme in the Pass East had said: &quot;Youzhou — born to be destroyed; if not destroyed, the people shall be extinguished.&quot; This was [Murong] Cui's birth name. Having held out against [Fu] Pi for a full year, the common people were nearly all dead.
#: '''Original:''' 慕容垂军人饥甚，多奔中山，幽、冀人相食。初，关东谣曰：&quot;幽州，生当灭。若不灭，百姓绝。&quot;（慕容）垂之本名。与（符）丕相持经年，百姓死几绝。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 114 &quot;Chronicles 14, Fu Jian II&quot; (《晋书·卷一百十四·载记第十四·苻坚下》)
## 385 CE: Famine in You and Ji Prefectures, ''Zizhi Tongjian''
##: '''English''': Yan and Qin having held out against each other for a full year, You and Ji prefectures suffered a Great Famine; people ate each other, and settlements lay desolate. Many of Yan's soldiers starved to death; the King of Yan, [Murong] Cui, forbade the people from raising silkworms and had them subsist on mulberry berries.
##: '''Original:''' 燕、秦相持經年，幽、冀大饑，人相食，邑落蕭條，燕之軍士多餓死，燕王（慕容）垂禁民養蠶，以桑椹為食。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 106 (《资治通鉴·卷一百零六》)
# 386 CE: Fu Deng's Army Eats the Slain, ''Book of Jin''
#: '''English''': [Fu] Deng, having succeeded Wei Ping, thenceforth held sole command of military campaigns. At this time drought brought widespread hunger, and the roads were lined with the starving dead. Whenever Deng won a battle and slew the enemy, he called it &quot;cooked meat,&quot; and said to his men: &quot;You fight in the morning and by evening are sated with flesh — why fear hunger!&quot; The troops followed his lead, eating the flesh of the slain, and were thereby well-fed and fit for battle.
#: '''Original:''' （苻）登既代卫平，遂专统征伐。是时岁旱众饥，道殣相望，登每战杀贼，名为熟食，谓军人曰：&quot;汝等朝战，暮便饱肉，何忧于饥！&quot;士众从之，啖死人肉，辄饱健能斗。 
#: '''Source:''' [[wikipedia:Book of Jin|''Book of Jin'']], Vol. 115 &quot;Chronicles 15, Fu Pi et al.&quot; (《晋书·卷一百十五·载记第十五·苻丕等》)
# 387 CE: Famine in Jiuquan, ''Book of Jin''
#: '''English''': Wang Mu seized Jiuquan by surprise and proclaimed himself General-in-Chief and Governor of Liangzhou. At this time grain prices soared; one dou fetched five hundred cash, people ate each other, and more than half perished.
#: '''Original:''' 王穆袭据酒泉，自称大将军、凉州牧。时谷价踊贵，斗直五百，人相食，死者太半。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 122 &quot;Chronicles 22, Lü Guang et al.&quot; (《晋书·卷一百二十二·载记第二十二·吕光等》)
# 387 CE: Famine in Liangzhou, ''Zizhi Tongjian''
#: '''English''': Great Famine in Liangzhou; one dou of rice fetched five hundred cash, people ate each other, and more than half perished.
#: '''Original:''' 涼州大饑，米斗直錢五百，人相食，死者太半。
#: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 112 (《资治通鉴·卷第一百一十二》)
# c. 399 CE: Sun En Rebellion, ''Song Shu''
#: '''English''': In this time all means of livelihood were exhausted and the weak and elderly were many; the eastern lands suffered famine, and people exchanged children to eat. 
#: '''Original:''' 时生业已尽，老弱甚多，东土饥荒，易子而食;
#: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 100 &quot;Biographies 60, Preface&quot; (《宋书·卷一百·列传第六十·自序》)
## c. 399 CE: Sun En Rebellion, ''Wei Shu''
##: '''English''': When [Sun] En raised his rebellion, all eight commanderies became a field of carnage. … The rebels' prohibitions went unheeded; they killed at will, and the number of officers and commoners slain was beyond reckoning. Some county magistrates were pickled and fed to their own wives and children; those who refused were dismembered. Such was their cruelty.
##: '''Original:''' （孙）恩既作乱，八郡尽为贼场，……贼等禁令不行，肆意杀戮，士庶死者不可胜计，或醢诸县令以食其妻子，不肯者辄支解之，其虐如此。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 96 &quot;Biographies 84, the Usurper Jin's Sima Rui et al.&quot; (《魏书·卷九十六·列传第八十四·僭晋司马叡等》)
# 401 CE, Longan 5: Omen of Famine and Usurpation, ''Book of Jin''
#: '''English''': Huan Xuan's memorial arrived, defying imperial intent and affronting the throne. Thereafter Xuan usurped the throne, threw the capital into disorder; there was a Great Famine, people ate each other, and the common people fled — all were fulfillments of these omens.
#: '''Original:''' 九月，桓玄表至，逆旨陵上。其后玄遂篡位，乱京都，大饥，人相食，百姓流亡，皆其应也。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 13 &quot;Treatises 3, Astronomy III&quot; (《晋书·卷十三·志第三·天文下》)
# 402 CE: Famine at Guzang, ''Book of Jin''
#: '''English''': Grain prices at Guzang soared; one dou fetched five thousand cash, people ate each other, and over a hundred thousand starved to death. The city gates were shut by day, and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the barbarians. [Lü] Long, fearing this would demoralize the populace, had them all buried alive, corpses piled up and filled the streets.
#: '''Original:''' 姑臧谷价踊贵，斗直钱五千文，人相食，饿死者十余万口。城门昼闭，樵采路绝，百姓请出城乞为夷虏奴婢者日有数百。隆惧沮动人情，尽坑之，于是积尸盈于衢路。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 122 &quot;Chronicles 22, Lü Guang et al.&quot; (《晋书·卷一百二十二·载记第二十二·吕光等》)
## 402 CE: Famine at Guzang, ''Wei Shu''
##: '''English''': Juqu Mengxun and Tufa Rutan attacked repeatedly, leaving the people of Hexi unable to farm to the west. Grain prices soared; one dou fetched five thousand cash, people ate each other, and over a thousand starved to death. The city gates of Guzang were shut by day and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the barbarians. [Lü] Long, fearing this would demoralize the populace, had them all buried alive.
##: '''Original:''' 沮渠蒙逊、秃发辱檀频来攻击，河西之民，不得农西，谷价涌贵，斗直钱五千文，人相食，饿死者千余口。姑臧城门昼闭，樵采路断，民请出城，乞为夷虏奴婢者，日有数百。隆恐沮动人情，尽坑之。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 95 &quot;Biographies 83, Liu Cong of the Xiongnu et al.&quot; (《魏书·卷九十五·列传第八十三·匈奴刘聪等》)
## 402 CE: Famine at Guzang, ''Zizhi Tongjian''
##: '''English''': Great Famine at Guzang; one dou of rice fetched five thousand cash, people ate each other, and over a hundred thousand starved to death. The city gates were shut by day, and the roads for gathering firewood were cut off. Each day several hundred commoners petitioned to leave the city and offer themselves as slaves to the Hu barbarians; Lü Long, loathing the effect on morale, had them all buried alive, corpses piled up and filled the roads.
##: '''Original:''' 姑臧大饥，米斗直钱五千，人相食，饥死者十馀万口。城门昼闭，樵采路绝，民请出城为胡虏奴婢者，日有数百，吕隆恶其沮动众心，尽坑之，积尸盈路。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']], Vol. 112 (《资治通鉴·卷一百一十二》)
# 402 CE: Astronomical Omen of Famine, ''Book of Jin''
#: '''English''': In the fourth month, on the day xinsi, the moon occluded Mercury. In the seventh month, Great Famine; people ate each other.
#: '''Original:''' 元兴元年四月辛丑，月奄辰星。七月，大饥，人相食。
#: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 12 &quot;Treatises 2, Astronomy II&quot; (《晋书·卷十二·志第二·天文中》)
## 402 CE: Famine in the Eastern Regions, ''Book of Jin(1)''
##: '''English''': In the seventh month of Yuanxing 1, Great Famine; people ate each other. Six or seven in ten east of the Zhe River died or fled; the population of Wu Commandery and Wuxing was halved, and tens of thousands more fled westward.
##: '''Original:''' 元兴元年七月，大饥，人相食。浙江以东流亡十六七，吴郡、吴兴户口减半，又流奔而西者万计。
##: '''Source:''' [[:w:Book of Jin|''Book of Jin'']], Vol. 13 &quot;Treatises 3, Astronomy III&quot; (《晋书·卷十三·志第三·天文下》)
## 402 CE: Famine in the Eastern Regions, ''Song Shu''
##: '''English''': In the seventh month [of Yuanxing 1], Great Famine; people ate each other. Six or seven in ten east of the Zhe River starved to death or fled; the population of Wu Commandery and Wuxing was halved, and tens of thousands more fled westward.
##: '''Original:''' （元兴元年）七月，大饥，人相食。浙江东饿死流亡十六七，吴郡、吴兴户口减半；又流奔而西者万计。
##: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 25 &quot;Treatises 15, Astronomy III&quot; (《宋书·卷二十五·志第十五·天文三》)
# 402 CE Kong Clan Distributes Grain, ''Song Shu''
#: '''English''': After the Sun En rebellion, the eastern lands suffered famine; people ate each other. The Kong clan distributed their household grain to relieve the neighbourhood, saving many lives; those who bore children thereafter named them Kong in gratitude.
#: '''Original:''' 及孙恩乱后，东土饥荒，人相食，孔氏散家粮以赈邑里，得活者甚众，生子皆以孔为名焉。
#: '''Source:''' [[:w:Song Shu|''Song Shu'']], Vol. 81 &quot;Biographies 41, Liu Xiuzhi et al.&quot; (《宋书·卷八十一·列传第四十一·刘秀之等》)
## 402 CE: Kong Clan Distributes Grain, ''Nan Shi''
##: '''English''': After the Sun En rebellion, the eastern lands suffered famine; people ate each other. The Kong clan distributed their household grain to relieve the neighbourhood, saving many lives; those who bore children thereafter named them Kong in gratitude.
##: '''Original:''' 孙恩乱后，东土饥荒，人相食，孔氏散家粮以振邑里，得活者甚众，生子皆以孔为名焉。
##: '''Source:''' [[:w:Nan Shi|''Nan Shi'']], Vol. 35 &quot;Biographies 25, Liu Zhan et al.&quot; (《南史·卷三十五·列传第二十五·刘湛等》)
# 409 CE: Cannibalism as Punishment for Regicide, ''Bei Shi''
#: '''English''': [Tuoba] Shao, together with several attendants and eunuchs, scaled the palace walls and violated the forbidden precinct. The Emperor [Daowu of Northern Wei, Tuoba Gui] started up in alarm, reached for his bow and sword but could not find them, and died suddenly. … The guards seized and delivered Shao. Thereupon Shao and his mother were put to death; the attending eunuchs and palace women who had acted as inner accomplices, numbering over ten, were executed. Those who had first laid hands upon the imperial person were carved alive and eaten by the assembled ministers on the main avenue south of the city.
#: '''Original:''' （拓跋）绍乃与帐下及宦者数人逾宫犯禁。帝（北魏道武皇帝拓跋珪）惊起，求弓刀不及，暴崩。……卫士执送绍，于是赐绍母子死，诛帐下阉官、宫人为内应者十数人。其先犯乘舆者，群臣于城南都街生脔食之。
#: '''Source:''' [[:w:Bei Shi|''Bei Shi'']], Vol. 16 &quot;Biographies 4, The Seven Princes of Daowu et al.&quot; (《北史·卷十六·列传第四·道武七王等》)
## 409 CE: Cannibalism as Punishment for Regicide, ''Wei Shu''
##: '''English''': The Supreme Ancestor (Taizong) arrived at the west of the city; the guards seized and delivered Shao. Thereupon Shao and his mother were put to death; the attending eunuchs and palace women who had acted as inner accomplices, numbering over ten, were executed. Those who had first laid hands upon the imperial person were carved alive and eaten by the assembled ministers on the main avenue south of the city.
##: '''Original:''' 太宗至城西，卫士执送绍。于是赐绍母子死，诛帐下阉官、宫人为内应者十数人，其先犯乘舆者，群臣于城南都街生脔割而食之。
##: '''Source:''' [[:w:Wei Shu|''Wei Shu'']], Vol. 16 &quot;Biographies 4, The Seven Princes of Daowu&quot; (《魏书·卷十六·列传第四·道武七王》)
## 409 CE: Cannibalism as Punishment for Regicide, ''Zizhi Tongjian''
##: '''English''': Those who had first laid hands upon the imperial person [Tuoba Gui] were carved and eaten by the assembled ministers.
##: '''Original:''' 其先犯乘舆（拓跋珪）者，群臣脔食之。
##: '''Source:''' [[:w:Zizhi Tongjian|''Zizhi Tongjian'']] (《资治通鉴》)


==Northern and Southern dynasties==

# 431 CE: Siege of Nan'an, ''Bei Shi''
#: '''English''': Helian Ding dispatched Wei Dai, Duke of Beiping, with ten thousand men to attack Nan'an. Within the city there was Great Famine; people ate each other.
#: '''Original:''' 赫连定遣其北平公韦代率众万人攻南安。城内大饥，人相食。
#: '''Source:''' [[:w:Bei Shi|Bei Shi]], Vol. 93 &quot;Biographies, 81: Pretenders and Vassals&quot; (《北史·卷九十三·列传第八十一·僭伪附庸》)
## 431 CE: Siege of Nan'an, ''Book of Wei''
##: '''English''': Helian Ding dispatched Wei Dai, Duke of Beiping, with ten thousand men to attack Nan'an. Within the city there was Great Famine; people ate each other.
##: '''Original:''' 赫连定遣其北平公韦代率众一万攻南安，城内大饥，人相食。
##: '''Source:''' [[:w:Wei Shu|Wei Shu]], Vol. 99 &quot;Biographies, 87: Zhang Shi, Governor of Liangzhou et al.&quot; (《魏书·卷九十九·列传第八十七·凉州牧张实等》)
## 431 CE: Siege of Nan'an, ''Zizhi Tongjian''
##: '''English''': The Xia ruler (Helian Ding) attacked and defeated the Qin general Yao Xian; thereupon he dispatched his uncle Wei Fa, Duke of Beiping, with ten thousand men to attack Nan'an. Within the city there was Great Famine; people ate each other.
##: '''Original:''' 夏主（赫连定）击秦将姚献，败之；遂遣其叔父北平公韦伐帅众一万攻南安。城中大饥，人相食。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 122 (《资治通鉴》卷122)
# Yuanjia Era: Medicinal Corpse, ''Yi Yuan''
#: '''English''': In the Yuanjia era, the Hu family of Yuzhang opened the tomb of [[:w:Marquis of Haihun | King Changyi]], and a man of Qingzhou opened the tomb of [[:w:Duke Xiang of Qi|Duke Xiang of Qi]]; both found golden hooks, whilst the corpses remained intact in the rocks. This may not be certain, yet the corpse of [[:w:Jing Fang|Jing Fang]] remained complete until the Yixi era; the flesh of such frozen corpses was fit for medicine, and soldiers carved and ate thereof.
#: '''Original:''' 元嘉中，豫章胡家奴開邑王冢，青州人開齊襄公冢，並得金鉤，而屍骸露在岩中儼然。茲亦未必有憑而然也，京房屍至義熙中猶完具，殭屍人肉堪為藥，軍士分割食之。
#: '''Source:''' [[:w:zh:异苑|Yi Yuan]] by Liu Jingshu (《异苑》)
# 441 CE: Siege of Jiuquan, ''Book of Song''
#: '''English''': In the seventh month, Tuoba Tao dispatched an army to besiege Jiuquan. In the tenth month, there was famine within the city and ten thousand people starved to death; Juqu Tianzhou killed his wife to feed the soldiers. When the food was exhausted, the city fell; Tianzhou was captured and taken to Pingcheng, where he was executed.
#: '''Original:''' 七月，拓跋焘遣军围酒泉。十月，城中饥，万余口皆饿死，（沮渠）天周杀妻以食战士；食尽，城乃陷，执天周至平城，杀之。
#: '''Source:''' [[:w:Song Shu|Song Shu]], Vol. 98 &quot;Biographies, 58: Di Hu&quot; (《宋书·卷九十八·列传第五十八·氐胡》)
## 441 CE: Siege of Jiuquan, ''Zizhi Tongjian''
##: '''English''': Food was exhausted within the city of Jiuquan and ten thousand people starved to death; Juqu Tianzhou killed his wife to feed the soldiers.
##: '''Original:''' 酒泉城中食尽，万馀口皆饿死，沮渠天周杀妻以食战士。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 123 (《资治通鉴》卷123)
# c. 450 CE: Qingzhou Famine, ''Book of Southern Qi''
#: '''English''': At the end of the Yuanjia era, there was famine in Qingzhou; people ate each other.
#: '''Original:''' 元嘉末，青州饥荒，人相食。
#: '''Source:''' [[:w:Nan Qi Shu|Book of Southern Qi]], Vol. 28 &quot;Biographies, 9: Cui Zushi et al.&quot; (《南齐书·卷二十八·列传第九·崔祖思等》)
## c. 450 CE: Qingzhou Famine, ''Nan Shi''
##: '''English''': At the end of the Yuanjia era, there was famine in Qingzhou; people ate each other. (Liu) Shanming had stored grain; he himself ate only thin porridge and opened his granaries to provide relief, whereby many in the village were saved. The people thereafter called his fields the &quot;Life-Sustaining Fields.&quot;
##: '''Original:''' 元嘉末，青州饥荒，人相食。（刘）善明家有积粟，躬食饘粥，开仓以救，乡里多获全济，百姓呼其家田为续命田。
##: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 49 &quot;Biographies, 39: Yu Gaozhi et al.&quot; (《南史·卷四十九·列传第三十九·庾杲之等》)
# 453 CE: Execution of Zhang Chaozhi, ''Song Shu''
#: '''English''': Zhang Chaozhi, hearing the troops had entered, fled to the old foundations of the He-dian hall and stopped at the site of the imperial bed, where he was killed by rebel soldiers. They cut open his intestines, gouged out his heart, and carved his flesh; the generals ate it raw and burned his skull.
#: '''Original:''' 张超之闻兵入，遂走至合殿故基，正于御床之所，为乱兵所杀。割肠刳心，脔剖其肉，诸将生啖之，焚其头骨。
#: '''Source:''' [[:w:Song Shu|Song Shu]], Vol. 99 &quot;Biographies, 59: Two Villains&quot; (《宋书·卷九十九·列传第五十九·二凶》)
## 453 CE: Execution of Zhang Chaozhi, ''Nan Shi''
##: '''English''': Zhang Chaozhi fled to the site of the imperial bed in the He-dian hall. He was killed by soldiers; they gouged his intestines and heart, carved his flesh, and the generals ate it raw. They burned his skull.
##: '''Original:''' 张超之走至合殿御床之所。为军士所杀，刳肠割心，诸将脔其肉，生啖之。焚其头骨。
##: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 14 &quot;Biographies, 4: Song Imperial Clan and Princes&quot; (《南史·卷十四·列传第四·宋宗室及诸王下》)
## 453 CE: Execution of Zhang Chaozhi, ''Zizhi Tongjian''
##: '''English''': Zhang Chaozhi fled to the site of the imperial bed in the He-dian hall. He was killed by soldiers; they gouged his intestines and heart, and the generals carved his flesh and ate it raw.
##: '''Original:''' 张超之走至合殿御床之所。为军士所杀，刳肠割心，诸将脔其肉，生啖之。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 127 (《资治通鉴》卷127)
# c. 454 CE: Liu Yong's Consumption of Scabs, ''Book of Song''
#: '''English''': Liu Yong had a passion for eating scabs, believing the taste resembled dried fish. He once visited Meng Lingxiu; Lingxiu had previously suffered from cautery sores, and the scabs had fallen upon the bed, whereupon Liu Yong took and ate them. Lingxiu was greatly alarmed. Liu Yong replied, &quot;It is my nature to love this.&quot; Lingxiu then stripped away all remaining scabs from his body to provide for Liu Yong. After Liu Yong departed, Lingxiu wrote to He Xu, saying, &quot;Liu Yong just looked at me and devoured me, until my whole body bled.&quot; In Nankang Commandery, some two hundred officials, regardless of whether they were guilty or innocent, were whipped in rotation so that the resulting scabs might constantly provide for his meals.
#: '''Original:''' （刘）邕所至嗜食疮痂，以为味似鳆鱼。尝诣孟灵休，灵休先患灸疮，疮痂落床上，因取食之。灵休大惊。答曰：“性之所嗜。”灵休疮痂未落者，悉褫取以饴邕。邕既去，灵休与何勖书曰：“刘邕向顾见啖，遂举体流血。”南康国吏二百许人，不问有罪无罪，递互与鞭，鞭疮痂常以给膳。
#: '''Source:''' [[:w:Book of Song|Book of Song]], Vol. 42 &quot;Biographies, 2: Liu Muzhi et al.&quot; (《宋书·卷四十二·列传第二·刘穆之等》)
## c. 454 CE: Liu Yong's Consumption of Scabs, ''Nan Shi''
##: '''English''': Liu Yong had a passion for eating scabs, believing the taste resembled abalone. He once visited Meng Lingxiu; Lingxiu had previously suffered from blistions caused by [[:w:Moxibustion|moxibustion]], and the scabs fell upon the bed, which Liu Yong took and ate. Lingxiu was greatly alarmed; he then stripped away all remaining scabs to provide for Liu Yong. After Liu Yong departed, Lingxiu wrote to He Xu, saying, &quot;Liu Yong just looked at me and devoured me, until my whole body bled.&quot; In Nankang Commandery, some two hundred officials, regardless of whether they were guilty or innocent, were whipped in rotation, and the scabs were constantly provided for his meals.
##: '''Original:''' （刘）邕性嗜食疮痂，以为味似鳆鱼。尝诣孟灵休，灵休先患灸疮，痂落在床，邕取食之。灵休大惊，痂未落者，悉褫取饴邕。邕去，灵休与何勖书曰：“刘邕向顾见啖，遂举体流血。”南康国吏二百许人，不问有罪无罪，递与鞭，疮痂常以给膳。
##: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 15 &quot;Biographies, 5: Liu Muzhi et al.&quot; (《南史·卷十五·列传第五·刘穆之等》)
# 465 CE: Mutilation of Wang Yigong, ''Nan Shi''
#: '''English''': The former deposed Emperor (Liu Ziye) was maddened and lawless. Wang Yigong and Liu Yuanjing conspired to depose him; the deposed Emperor led the Yulin guards to their residences and slew them, along with their four sons. He cut and severed the limbs of Wang Yigong, split open his abdomen and stomach, and plucked out his eyes to soak them in honey, calling them &quot;Ghost-Eye [[:w:Zongzi|Zongzi]].&quot;
#: '''Original:''' 前废帝（刘子业）狂悖无道，（王）义恭、（柳）元景谋欲废立，废帝率羽林兵于第害之，并其四子。断析义恭支体，分裂腹胃，挑取眼睛以蜜渍之，以为鬼目粽。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 13 &quot;Biographies, 3: Imperial Clan and Various Princes&quot; (《南史·卷十三·列传第三·宋宗室及诸王上》)
## 465 CE: Mutilation of Wang Yigong, ''Zizhi Tongjian''
##: '''English''': The Emperor (the former deposed Emperor of the Southern Song, Liu Ziye) personally led the Yulin guards to attack Wang Yigong and slew him, along with his four sons. He severed the limbs of Wang Yigong, split open his intestines and stomach, plucked out his eyes, and soaked them in honey, calling them &quot;Ghost-Eye Zongzi.&quot;
##: '''Original:''' 帝（南朝宋前废帝刘子业）自帅羽林兵讨（王）义恭，杀之，并其四子。断绝义恭支体，分裂肠胃，挑取眼睛，以蜜渍之，谓之“鬼目粽”。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 130 (《资治通鉴》卷130)
# 498 CE: Cannibalism of Huang Yaoqi, ''Book of Southern Qi''
#: '''English''': The barbarian forces pursued and captured Huang Yaoqi; Wang Su recruited men to carve up and eat his flesh.
#: '''Original:''' 虏追军获（黄）瑶起，王肃募人脔食其肉。
#: '''Source:''' [[:w:Book of Southern Qi|Book of Southern Qi]], Vol. 57 &quot;Biographies, 38: Wei Barbarians&quot; (《南齐书·卷五十七·列传第三十八·魏虏》)
## 498 CE: Cannibalism of Huang Yaoqi, ''Nan Shi''
##: '''English''': Wang Chen's brothers, Su and Bing, both fled to Wei; later they captured Huang Yaoqi, carved him up, and ate him.
##: '''Original:''' （王）琛弟肃、秉并奔魏，后得黄瑶起脔食之。
##: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 23 &quot;Biographies, 13: Wang Dan et al.&quot; (《南史·卷二十三·列传第十三·王诞等》)
## 498 CE: Cannibalism of Huang Yaoqi, ''Zizhi Tongjian''
##: '''English''': Huang Yaoqi was captured by Wei; the Lord of Wei bestowed him upon Wang Su, who carved him up and ate him.
##: '''Original:''' （黄）瑶起为魏所获，魏主以赐王肃，肃脔而食之。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 141 (《资治通鉴》卷141)
# 499 CE: Siege of Maquan City, ''Book of Southern Qi''
#: '''English''': In the first year of Yongyuan, Chen Xianda supervised General Cui Huijing and forty thousand troops to besiege Maquan City in Nanxiang, three hundred li from Xiangyang, attacking for forty days. The barbarians' food was exhausted; they ate the flesh of dead men and tree bark.
#: '''Original:''' 永元元年，（陈）显达督平北将军崔慧景众军四万，围南乡堺马圈城，去襄阳三百里，攻之四十日。虏食尽，啖死人肉及树皮。
#: '''Source:''' [[:w:Book of Southern Qi|Book of Southern Qi]], Vol. 26 &quot;Biographies, 7: Wang Jingze, Chen Xianda&quot; (《南齐书·卷二十六·列传第七·王敬则 陈显达》)
## 499 CE: Siege of Maquan City, ''Nan Shi''
##: '''English''': In the first year of Yongyuan, Chen Xianda supervised General Cui Huijing and forty thousand troops to besiege Maquan City in Nanxiang, three hundred li from Xiangyang. They attacked for forty days; the Wei army's food was exhausted, and they ate the flesh of dead men and tree bark.
##: '''Original:''' 永元元年，（陈）显达督平北将军崔慧景众军四万，围南乡界马圈城，去襄阳三百里。攻之四十日，魏军食尽，啖死人肉及树皮。
##: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 45 &quot;Biographies, 35: Wang Jingze et al.&quot; (《南史·卷四十五·列传第三十五·王敬则等》)
## 499 CE: Siege of Maquan City, ''Zizhi Tongjian''
##: '''English''': Chen Xianda fought Wei Yuanying and repeatedly defeated him. He sieged Maquan City for forty days; the food within the city was exhausted, and they ate the flesh of dead men and tree bark.
##: '''Original:''' 陈显达与魏元英战，屡破之。攻马圈城四十日，城中食尽，啖死人肉及树皮。
##: '''Source:''' [[:w:Zizhi Tongjian|Zizhi Tongjian]], Vol. 142 (《资治通鉴》卷142)
# 502 CE: Aftermath of Sun Wenming's Rebellion, Nan Shi
#: '''English''': At that time, the remnants of the Eastern Tyrant, including Sun Wenming and others, rebelled. Zhang Hongce jumped over a wall to hide in the dragon stables, where he encountered rebels and was thereupon slain. The government army captured Sun Wenming and executed him in the East Market; the kinsmen of the Zhang family carved him up and ate him.
#: '''Original:''' 时东昏余党孙文明等……作乱，……（张）弘策踰垣匿于龙厩，遇贼见害。……官军捕文明斩于东市，张氏亲属脔食之。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 56 &quot;Biographies, 46: Zhang Hongce et al.&quot; (《南史·卷五十六·列传第四十六·张弘策等》)
#: '''Original:''' 时东昏余党孙文明等……作乱，……（张）弘策踰垣匿于龙厩，遇贼见害。……官军捕文明斩于东市，张氏亲属脔食之。
#: '''Source:''' [[:w:Nan Shi|Nan Shi]], Vol. 56 &quot;Biographies, 46: Zhang Hongce et al.&quot; (《南史·卷五十六·列传第四十六·张弘策等》)
#  503年: 成都城中食尽，升米三千，人相食。（《资治通鉴》卷145）
#   约525年: 大将军萧宝夤西讨，德广为行台郎，募众而征，战捷，乃手刃仇人，啖其肝肺。（《北史·卷一百·序传第八十八》㉕*）
#   525年: 山胡刘蠡升自云圣术，胡人信之，咸相影附，旬日之间，逆徒还振。……先是官粟贷民。未及收聚，仍值寇乱。至是（汾州）城民大饥，人相食。贼知仓库空虚，攻围日甚，死者十三四。（裴）良以饥窘，因与城人奔赴西河。（《魏书·卷六十九·列传第五十七·崔休等》㉕*）
#  529年: 于是（元颢）斩（杨）昱下统帅三十七人，皆令蜀兵刳腹取心食之。（《北史·卷四十一·列传第二十九·杨播等》㉕*）&lt;p&gt;于是（元颢）斩（杨）昱下统帅三十七人，皆令蜀兵刳腹取心食之。（《魏书·卷五十八·列传第四十六·杨播》㉕）&lt;/p&gt;&lt;p&gt;于是（元颢）斩（杨）昱所部统帅三十七人，皆刳心而食之。 （《资治通鉴》卷153）&lt;/p&gt;
#   约532年：（北方）于时年凶，人多相食，昕勤恤人隐，多所全济。（《北史·卷二十四·列传第十二·崔逞等》㉕*）
#   约533年: 中大通四年，（梁武帝萧衍）特封（萧正德）临贺郡王。后为丹阳尹，坐所部多劫盗，复为有司所奏，去职。出为南兖州，在任苛刻，人不堪命。广陵沃壤，遂为之荒，至人相食啖。（《南史·卷五十一·列传第四十一·梁宗室上》㉕*）
#  536年: 是岁，关中大饥，人相食，死者十七八。（《北史·卷五·魏本纪第五》㉕*）&lt;p&gt; （西）魏关中大饥，人相食，死者什七八。  （《资治通鉴》卷157）&lt;/p&gt;
#  548年: 景食石头常平仓既尽，便掠居人，尔后米一升七八万钱，人相食，有食其子者。又筑土山，不限贵贱，昼夜不息，乱加殴棰，疲羸者因杀以填山，号哭之声动天地。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;石头常平诸仓既尽，（侯景）军中乏食；乃纵士卒掠夺民米及金帛子女。是后米一升直七八万钱，人相食，饿死者什五六。 （《资治通鉴》卷161）&lt;/p&gt;
#   548年: 鄱阳世子嗣、永安侯确、羊鸦仁、李迁仕、樊文皎率众度淮，攻破贼（侯景）东府城前栅，遂营于青溪水东。（侯）景遣其仪同宋子仙缘水西立栅以相拒。景食稍尽，人相食者十五六。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;景遣其仪同宋子仙顿南平王第，缘水西立栅相拒。景食稍尽，至是米斛数十万，人相食者十五六。（《梁书·卷五十六·列传第五十·侯景》㉕）&lt;/p&gt;
#   549年， [[:w:梁武帝|梁武帝]]太清三年：贼（侯景）之始至，（建邺）城中才得固守，平荡之事，期望援军。既而中外断绝，……军人屠马于殿省间鬻之，杂以人肉，食者必病。（《南史·卷八十·列传第七十·贼臣》㉕*）&lt;p&gt;（萧）衍城内大饥，人相食，米一斗八十万，皆以人肉杂牛马而卖之。（《魏书·卷九十八·列传第八十六·岛夷萧道成等》㉕）&lt;/p&gt;&lt;p&gt;（梁）军人屠马于殿省间，杂以人肉，食者必病。 （《资治通鉴》卷162）&lt;/p&gt;
#  549年: 自（侯）景作乱，（建康）道路断绝，数月之间，人至相食，犹不免饿死，存者百无一二。贵戚、豪族皆自出采稆，填委沟壑，不可胜纪。  （《资治通鉴》卷162）
#   549年，梁太清三年：是月（七月），九江大饥，人相食十四五。（《梁书·卷四·本纪第四·简文帝》㉕*）&lt;p&gt;九江大饥，人相食者十四五。（《南史·卷八·梁本纪下第八》㉕）&lt;/p&gt;&lt;p&gt;是年，帝为侯景所幽，崩。七月，九江大饥，人相食十四五。（《隋书·卷二十一·志第十六·天文下》㉕）&lt;/p&gt;
#  550: 值梁室丧乱，（姚察）于金陵随二亲还乡里。时东土兵荒，人饥相食，告籴无处，察家口既多，并采野蔬自给。（《陈书· 卷二十七·列传第二十一·江总 姚察》㉕*）&lt;p&gt;自晋氏度江，三吴最为富庶，贡赋商旅，皆出其地。及侯景之乱，掠金帛既尽，乃掠人而食之，或卖于北境，遗民殆尽矣。  （《资治通鉴》卷163）&lt;/p&gt;
#   550年，梁大宝元年：自春迄夏，大饥，人相食，京师尤甚。（《梁书·卷四·本纪第四·简文帝》㉕*）&lt;p&gt;自春迄夏大旱，人相食，都下尤甚。（《南史·卷八·梁本纪下第八》㉕）&lt;/p&gt;
#   552年:（侯）景不能制，乃与腹心数十人单舸走，推堕二子于水，自沪渎入海。至壶豆洲，前太子舍人羊鲲杀之，送尸于王僧辩，传首西台，曝尸于建康市。百姓争取屠脍啖食，焚骨扬灰。（《梁书·卷五十六·列传第五十·侯景》㉕*）&lt;p&gt;及（侯）景死，僧辩截其二手送齐文宣，传首江陵，果以盐五斗置腹中，送于建康，暴之于市。百姓争取屠脍羹食皆尽，并溧阳主亦预食例。景焚骨扬灰，曾罹其祸者，乃以灰和酒饮之。（《南史·卷八十·列传第七十·贼臣》㉕）&lt;/p&gt;&lt;p&gt;既斩侯景，烹尸于建业市，百姓食之，至于肉尽龁骨，传首荆州，悬于都街。（《北齐书· 卷四十五·列传第三十七·文苑》㉕）&lt;/p&gt;&lt;p&gt;僧辩传（侯景）首江陵，截其手，使谢葳蕤送于齐；暴景尸于市，士民争取食之，并骨皆尽；溧阳公主亦预食焉。 （《资治通鉴》卷164）&lt;/p&gt;
#   552年: 王伟，陈留人。少有才学，景之表、启、书、檄，皆其所制。景既得志，规摹篡夺，皆伟之谋。及囚送江陵，烹于市，百姓有遭其毒者，并割炙食之。（《梁书·卷五十六·列传第五十·侯景》㉕*）
#  553年: （萧）圆照更无所言，唯云计误。并命绝食于狱，齿臂啖之，十三日死，天下闻而悲之。（《南史·卷五十三·列传第四十三·梁武帝诸子》㉕*）&lt;p&gt;上（梁元帝萧绎）并命（萧圆正）绝食于狱，至啮臂啖之，十三日而死，远近闻而悲之。 （《资治通鉴》卷165）&lt;/p&gt;
#  《南史》毗骞：“国法刑人，并于王前啖其肉。”“国内不受估客，往者亦杀而食之。”
#  554年: 五年春正月癸丑，帝（北齐文宣帝高洋）讨山胡大破之。男子十二已上皆斩，女子及幼弱以赏军。遂平石楼。石楼绝险，自魏代所不能至。于是远近山胡，莫不慑伏。是役也，有都督战伤，其什长路晖礼不能救，帝命刳其五藏，使九人分食之，肉及秽恶皆尽。自是始行威虐。（《北史·卷七·齐本纪中第七》㉕*）&lt;p&gt;有都督战伤，其什长路晖礼不能救，帝（北齐文宣帝高洋）命刳其五藏，令九人食之，肉及秽恶皆尽。（《资治通鉴》卷165）&lt;/p&gt;
#   555年: 众推（慕容）俨，遂遣镇郢城。……（侯）瑱、（任）约又并力围城。唯煮槐楮叶并纻根、水荭、葛、艾等及靴、皮带、筋角等食之。人死，即火别分食，唯留骸骨。俨犹信赏必罚，分甘同苦。自正月至六月，人无异志。（《北史·卷五十三·列传第四十一·万俟普等》㉕*）
#   约555年-560年: 自（天保）六年之后，帝（北齐文宣帝高洋）遂以功业自矜，恣行酷暴，昏狂酗醟，任情喜怒。为大镬、长锯、剉碓之属，并陈于庭，意有不快，则手自屠裂，或命左右脔啖，以逞其意。（《隋书·卷二十五·志第二十·刑法》㉕*）
#  流求国，居海岛，当建安郡东。水行五日而至。……国人好相攻击，……两军相当，勇者三五人出前跳噪，交言相骂，因相击射。如其不胜，一军皆走，遣人致谢，即共和解。收取斗死者聚食之，仍以髑髅将向王所，王则赐之以冠，便为队帅。……其南境风俗少异，人有死者，邑里共食之。（《北史·卷九十四·列传第八十二·高丽等》㉕*）&lt;p&gt;流求国，……南境风俗少异，人有死者，邑里共食之。（《隋书·卷八十一·列传第四十六·东夷》㉕）&lt;/p&gt;
#   獠者，盖南蛮之别种，自汉中达于邛、笮，川洞之间，所在皆有。……性同禽兽，至于忿怒，父子不相避，唯手有兵刃者先杀之。……若报怨相攻击，必杀而食之；（《北史·卷九十五·列传第八十三·蛮 獠 等》㉕*）
#   顿逊之外，大海洲中，又有毗骞国，去扶南八千里。……国法刑罪人，并于王前啖其肉。国内不受估客，有往者亦杀而啖之，是以商旅不敢至。（《梁书·卷五十四·列传第四十八·诸夷》㉕*）&lt;p&gt;又有毗骞国，去扶南八千里。……国法刑人，并于王前啖其肉。国内不受估客，有往者亦杀而啖之，是以商旅不敢至。（《南史· 卷七十八·列传第六十八·夷貊上》㉕）&lt;/p&gt;

==隋==
#   590年: 时江南州县又论言欲徙之入关，远近惊骇。饶州吴世华起兵为乱，生脔县令，啖其肉。（《北史·卷六十三·列传第五十一·周惠达等》㉕*）
#   隋文帝开皇年间（581-600年）：（杨武通）与周法尚讨嘉州叛獠，……贼知其孤军无援，倾部落而至。武通转斗数百里，为贼所拒，四面路绝。武通轻骑挑战，坠马，为贼所执，杀而啖之。（《北史·卷七十三·列传第六十一·梁士彦等》㉕*）&lt;p&gt;（杨）武通轻骑接战，坠马，为贼所执，杀而啖之。（《隋书·卷五十三·列传第十八·达奚长儒》㉕）&lt;/p&gt;
#   隋文帝开皇年间（581-600年）：郡中士女，号哭于路，诸郡惊骇，各奏其（王文同）事。帝闻大怒，遣使者违奚善意驰锁之（王文同），斩于河间，以谢百姓。仇人剖其棺，脔其肉啖之，斯须咸尽。（《北史·卷八十七·列传第七十五·酷吏》㉕*）&lt;p&gt;郡中士女号哭于路，诸郡惊骇，各奏其（王文同）事。帝闻而大怒，遣使者达奚善意驰锁之，斩于河间，以谢百姓，仇人剖其棺，脔其肉而啖之，斯须咸尽。（《隋书·卷七十四·列传第三十九·酷吏》㉕）&lt;/p&gt;
#   隋炀帝时代（604年-618年在位）中期：六军不息，百役繁兴；行者不归，居者失业；人饥相食，邑落为墟，上弗之恤也。（《北史·卷十二·隋本纪下第十二》㉕*）&lt;p&gt;六军不息，百役繁兴，行者不归，居者失业。人饥相食，邑落为墟，上不之恤也。（《隋书·卷四·帝纪第四·炀帝下》㉕）&lt;/p&gt;
#   613年: 及杨玄感反，帝（隋炀帝杨广）诛之，罪及九族。其尤重者，行轘裂枭首之刑。或磔而射之。命公卿已下，脔啖其肉。（《隋书·卷二十五·志第二十·刑法》㉕*）
#  614年:明年，（隋炀）帝复东征，高丽请和，遂送（斛斯）政。锁至京师以告庙，左翊卫大将军宇文述请变常法行刑，帝许之。以出金光门，缚之于柱，公卿百僚，并亲击射。脔其肉，多有啖者，然后烹焚，扬其骨灰。（《北史·卷四十九·列传第三十七·朱瑞等》㉕*）&lt;p&gt;（隋炀）帝复东征，高丽请降，求执送（斛斯）政。帝许之，遂锁政而还。至京师，以政告庙，左翊卫大将军字文述奏曰：“斛斯政之罪，天地所不容，人神所同忿。若同常刑，贼臣逆子何以惩肃？请变常法。”帝许之。于是将政出金光门，缚政于柱，公卿百僚并亲击射，脔割其肉，多有啖者。啖后烹煮，收其余骨，焚而扬之。（《隋书·卷七十·列传第三十五·杨玄感》㉕）&lt;/p&gt;&lt;p&gt;十一月，丙申，杀斛斯政于金光门外，如杨积善之法，仍烹其肉，使百官啖之，佞者或啖之至饱，收其馀骨，焚而扬之。 （《资治通鉴》卷182）&lt;/p&gt;
#   隋炀帝时代（604年-618年在位）后期：民外为盗贼所掠，内为郡县所赋，生计无遗；加之饥馑无食，民始采树皮叶，或捣穢为末，或煮土而食之，诸物皆尽，乃自相食。而官食犹充牣，吏皆畏法，莫敢振救。 （《资治通鉴》卷183）&lt;p&gt;相聚雚蒲，猬毛而起。大则跨州连郡，称帝称王；小则千百为群，攻城剽邑。流血成川泽，死人如乱麻；炊者不及析骸，食者不遑易子。（《北史·卷十二·隋本纪下第十二》㉕*）&lt;/p&gt;&lt;p&gt;俄而玄感肇黎阳之乱，匈奴有雁门之围，天子方弃中土，远之扬越。奸宄乘衅，强弱相陵，关梁闭而不通，皇舆往而不反。加之以师旅，因之以饥馑，流离道路，转死沟壑，十八九焉。于是相聚萑蒲，蝟毛而起，大则跨州连郡，称帝称王，小则千百为群，攻城剽邑，流血成川泽，死人如乱麻，炊者不及析骸，食者不遑易子。（《隋书·卷四·帝纪第四·炀帝下》㉕）&lt;/p&gt;&lt;p&gt;自燕赵跨于齐韩，江淮入于襄邓，东周洛邑之地，西秦陇山之右，僭伪交侵，盗贼充斥。宫观鞠为茂草，乡亭绝其烟火，人相啖食，十而四五。（《隋书·卷二十四·志第十九·食货》㉕）&lt;/p&gt;&lt;p&gt;是时百姓废业，屯集城堡，无以自给。然所在仓库，犹大充爨，吏皆惧法，莫肯赈救，由是益困。初皆剥树皮以食之，渐及于叶，皮叶皆尽，乃煮土或捣稿为末而食之。其后人乃相食。（《隋书·卷二十四·志第十九·食货》㉕）&lt;/p&gt;
#  616: 吏立木于市，悬其（张金称）头，张其手足，令仇家割食之；未死间，歌讴不辍。（《资治通鉴》卷183）
#  617年，大业十三年四月：（薛仁杲）所至多杀人，纳其妻妾。获庾信子立，怒其不降，磔于猛火之上，渐割以啖军士。（《旧唐书·卷五十五·列传第五·薛举等》㉕*）&lt;p&gt;（薛仁杲）尝得庾信子立，怒其不降，砾之火，渐割以啖士。（《新唐书·卷八十六·列传第十一 薛李二刘高徐》㉕）&lt;/p&gt;&lt;p&gt;（薛仁杲）尝获庾信子立，怒其不降，磔于火上，稍割以啖军士。”（《资治通鉴》卷183）&lt;/p&gt;
#  618: :（屈突）通引兵南遁，置（尧）君素领河东通守。……后颇得江都倾覆消息，又粮尽，男女相食，众心离骇。（《北史·卷八十五·列传第七十三·节义》㉕*）&lt;p&gt;时百姓苦隋日久，及逢义举，人有息肩之望。然君素善于统领，下不能叛。岁余，颇得外生口，城中微知江都倾覆。又粮食乏绝，人不聊生，男女相食，众心离骇。（《隋书·卷七十一·列传第三十六·诚节》㉕）&lt;/p&gt;&lt;p&gt;隋将尧君素守河东，上遣吕绍宗、韦义节、独孤怀恩相继攻之，俱不下。……久之，仓粟尽，人相食；（《资治通鉴》卷184）&lt;/p&gt;
#  618: （李轨）征兵筑台以候玉女，多所糜费，百姓患之。又属年饥，人相食，轨倾家赈之，私家罄尽，不能周遍。（谢统师等）乃诟珍曰：“百姓饿者自是弱人，勇壮之士终不肯困，国家仓粟须备不虞，岂可散之以供小弱？仆射苟悦人情，殊非国计。”轨以为然，由是士庶怨愤，多欲叛之。（《旧唐书·卷五十五·列传第五 薛举等》㉕*）&lt;p&gt;有胡巫妄曰：“上帝将遣玉女从天来。”（李轨）遂召兵筑台以候女，多所糜损。属荐饥，人相食，轨毁家赀赈之，不能给，议发仓粟，曹珍亦劝之。谢统师等故隋官，内不附，每引结群胡排其用事臣，因是欲离沮其众，乃廷诘珍曰：“百姓饥死皆弱不足事者，壮勇士终不肯困。且储禀以备不虞，岂宜妄散惠孱小乎？仆射苟附下，非国计。”轨曰：“善。”乃闭粟。下益怨，多欲叛去。（《新唐书·卷八十六·列传第十一·薛李二刘高徐》㉕） &lt;/p&gt;&lt;p&gt;有胡巫谓（李）轨曰：“上帝当遣玉女自天而降。”轨信之，发民筑台以候玉女，劳费甚广。河右饥，人相食，轨倾家财以赈之；不足，欲发仓粟，召群臣议之。曹珍等皆曰：“国以民为本，岂可爱仓粟而坐视其死乎！”谢统师等皆故隋官，心终不服，密与群胡为党，排轨故人，乃诟珍曰：“百姓饿者自是羸弱，勇壮之士终不至此。国家仓粟以备不虞，岂可散之以饲羸弱！仆射苟悦人情，不为国计，非忠臣也。”轨以为然，由是士民离怨。 （《资治通鉴》卷186）&lt;/p&gt;
#  619年：（朱）粲所克州县，皆发其藏粟以充食，迁徙无常，去辄焚余赀，毁城郭，又不务稼穑，以劫掠为业。于是百姓大馁，死者如积，人多相食。军中罄竭，无所虏掠，乃取婴儿蒸而啖之，因令军士曰：“食之美者，宁过于人肉乎！但令他国有人，我何所虑？”即勒所部，有略得妇人小儿皆烹之，分给军士，乃税诸城堡，取小弱男女以益兵粮。隋著作佐郎陆从典、通事舍人颜愍楚因谴左迁，并在南阳，粲悉引之为宾客，后遭饥馁，合家为贼所啖。（《旧唐书·卷五十六·列传第六·萧铣等》㉕*）&lt;p&gt;粲所克州县皆发藏粟以食，迁徙无常，去辄燔廥聚，毁城郭，不务稼穑，专以劫为资。于是人大馁，死者系路，其军亦匮，乃掠小儿烝食之。戒其徒曰：“味之珍宁有加人者？弟使佗国有人，我恤无储哉！”勒所部略妇人孺儿分烹之，又税诸城细弱以益粮。隋著作佐郎陆从典、通事舍人颜愍楚谪南阳，粲初引为宾客，后尽食两家。俄而诸城惧，皆逃散。（《新唐书·卷八十七·列传第十二·萧辅沈李梁》㉕）&lt;/p&gt;&lt;p&gt;朱粲有众二十万，剽掠汉、淮之间，迁徙无常，攻破州县，食其积粟未尽，复他适，将去，悉焚其余资；又不务稼穑，民馁死者如积。粲无可复掠，军中乏食，乃教士卒烹妇人、婴儿啖之，曰：“肉之美者无过于人，但使他国有人，何忧于馁！”隋著作佐郎陆从典、通事舍人颜愍楚，谪官在南阳，粲初引为宾客，其后无食，阖家皆为所啖。愍楚，之推之子也。又税诸城堡细弱以供军食，诸城堡相帅叛之。”（《资治通鉴》）&lt;/p&gt;&lt;p&gt;“隋末荒亂，狂賊[[:w:朱粲|朱粲]]起於襄、鄧間，歲飢，米斛萬錢，亦無得處，人民相食。粲乃驅男女小大仰一大銅鐘，可二百石，煮人肉以矮賊。生靈殲於此矣。”，朱粲竟說：“食之美者，寧過於人肉乎！”（唐·[[:w:張鷟|張鷟]]《朝野僉載》）&lt;/p&gt;
#  619年: （段）确醉，戏（朱）粲曰：“君脍人多矣，若为味？”粲曰：“啖嗜酒人，正似糟豚。”确悸，骂曰：“狂贼，归朝乃一奴耳，复得噬人乎？”粲惧，收确于坐，并从者数十悉饔之，以飨左右。遂屠菊潭，奔王世充，署龙骧大将军。东都平，斩洛水上。士庶竞掷瓦砾击其尸，须臾若冢。（《新唐书·卷八十七·列传第十二·萧辅沈李梁》㉕*）&lt;p&gt;（段确）乘醉侮（朱）粲曰：“闻卿好啖人，人作何味？”粲曰：“啖醉人正如糟藏彘肉。”确怒，骂曰：“狂贼入朝，为一头奴耳，复得啖人乎！”粲于座收确及从者数十人，悉烹之，以啖左右。（《资治通鉴》卷187）&lt;/p&gt;
#  隋末的[[:w:诸葛昂|诸葛昂]]與[[:w:高瓒|高瓒]]嗜食人肉。高瓒將双胞胎小孩杀掉，頭顱、手和腳分別裝在盤子裏，做成“双子宴”，與诸葛昂一起享用；诸葛昂则把自己的爱妾蒸熟，擺成盤腿打坐的姿勢，臉上重新塗好脂粉，諸葛昂親手撕她大腿上的肉請高瓒吃。（《[[:w:唐人说荟|唐人说荟]]》卷五，引张骞《耳目记》）

==唐==
安史之乱期间，张巡固守城池，城中人相食，张巡杀妾以飨将士，对于张巡以食人为代价的守土之功是否应该奖励，出现了一次伦理学的辩论，历代不息，《柏杨白话版资治通鉴》收集了若干历史上争论的意见。

黄巢之乱的时候，几支反叛军队成规模地常规性地以人为食，黄巢军“掠人为粮，生投于碓硙，并骨食之，号给粮之处曰‘舂磨寨’”，秦宗权军“啖人为储，军士四出，则盐尸而从”，李罕之军“不耕稼，专以剽掠为资，啖人为粮”。真是惨烈之甚。

唐朝陈藏器写的《本草拾遗》写人肉可以治病，这应该不是他的发明，而只是民间认知的一种总结，可能只是太多不得已的饥荒食人造成一种认知扭曲，但又反过来理性化了食人，到宋朝的时候，割肉疗亲开始出现。

#   621年，[[:w:唐高祖|唐高祖]]武德四年：（王）世充屯兵不散，仓粟日尽，城中人相食。或握土置瓮中，用水淘汰，沙石沉下，取其上浮泥，投以米屑，作饼饵而食之，人皆体肿而脚弱，枕倚于道路。其尚书郎卢君业、郭子高等皆死于沟壑。（《旧唐书·卷五十四·列传第四 王世充 窦建德》㉕*）&lt;p&gt;王（李世民）傅城，堑而守之。（王）世充粮且尽，人相食，至以水汨泥去砾，取浮土糅米屑为饼。民病肿股弱，相藉倚道上，其尚书郎卢君业、郭子高等皆饿死。御史大夫郑颋丐为浮屠，世充恶其言，杀之。（《新唐书·卷八十五·列传第十 王窦》㉕）&lt;/p&gt;
#621年: （单雄信）临将就戮，（李世）勣对之号恸，割股肉以啖之，曰：“生死永诀，此肉同归于土矣。”（《旧唐书·卷六十七·列传第十七·李靖等》㉕*）&lt;p&gt;（李世勣）乃割股肉以啖（单）雄信，曰：“使此肉随兄为土，庶几犹不负昔誓也！”（《资治通鉴》卷189）&lt;/p&gt;
#   627年: （王）君操密袖白刃刺杀之（杀父仇人李君则），刳腹取其心肝，啖食立尽，诣刺史具自陈告。（《旧唐书·卷一百八十八·列传第一百三十八·孝友》㉕*）
#   643年，[[:w:唐太宗|唐太宗]]贞观十七年: 贞观末，（刘兰）以谋反腰斩。右骁卫大将军丘行恭探其心肝而食之，太宗闻而召行恭让之曰：“典刑自有常科，何至于此！必若食逆者心肝而为忠孝，则刘兰之心为太子诸王所食，岂至卿邪？”行恭无以答。（《旧唐书·卷六十九·列传第十九·侯君集等》㉕*）&lt;p&gt;鄠尉[[:w:游文芝|游文芝]]告代州都督[[:w:劉蘭成|劉蘭成]]谋反，戊申，兰成坐[[:w:腰斩|腰斩]]。右武候将军[[:w:丘行恭|丘行恭]]，探兰成心肝食之。上（唐太宗）闻而让之曰：兰成谋反，国有常刑，何至如此！若以为忠孝，则太子诸王先食之矣，岂至卿耶？行恭惭而拜谢。（《资治通鉴》卷196）&lt;/p&gt;
#   约650年：周智寿者，雍州同官人。其父永徽初被族人安吉所害。智寿及弟智爽乃候安吉于途，击杀之。兄弟相率归罪于县，争为谋首，官司经数年不能决。乡人或证智爽先谋，竟伏诛。临刑神色自若，顾谓市人曰：“父仇已报，死亦何恨！”智寿顿绝衢路，流血遍体。又收智爽尸，舐取智爽血，食之皆尽，见者莫不伤焉。（《旧唐书·卷一百八十八·列传第一百三十八·孝友》㉕*）
#   662年: （郑）仁泰选骑万四千卷甲驰，绝大漠，至仙萼河，不见虏，粮尽还。人饥相食，比入塞，余兵才二十之一。（《新唐书·卷一百一十一·列传第三十六·郭二张三王苏薛程唐》㉕*)&lt;p&gt;（郑）仁泰将轻骑万四千，倍道赴之，遂逾大碛，至仙萼河，不见虏，粮尽而还。值大雪，士卒饥冻，弃捐甲兵，杀马食之，马尽，人自相食，比入塞，馀兵才八百人。（《资治通鉴》卷200）&lt;/p&gt;
#   682年，[[:w:唐高宗|唐高宗]]永淳元年：关中先水后早蝗，继以疾疫，米斗四百，两京间死者相枕于路，人相食。”（《资治通鉴》卷203）&lt;p&gt;六月，关中初雨，麦苗涝损，后旱，京兆、岐、陇螟蝗食苗并尽，加以民多疫疠，死者枕藉于路，诏所在官司埋瘗。京师人相食，寇盗纵横。（《旧唐书·卷五本纪第五·高宗下》㉕*）&lt;/p&gt;&lt;p&gt;永淳中，为雍州长史。时关中大饥，人相食，盗贼纵横。（《旧唐书·卷七十五·列传第二十五·苏世长等》㉕）&lt;/p&gt;&lt;p&gt;是月，大蝗，人相食。（《新唐书·卷三·本纪第三·高宗》㉕）&lt;/p&gt;&lt;p&gt;永淳元年，关中及山南州二十六饥，京师人相食。（《新唐书·卷三十五·志第二十五》㉕）&lt;/p&gt;&lt;p&gt;（良嗣）徙雍州。时关内饥，人相食，良嗣政上严，每盗发，三日内必擒，号称神明。（《新唐书·卷一百三·列传第二十八·苏世长等》㉕）&lt;/p&gt;
#   约684年: 王友贞，怀州河内人也。父知敬，则天时麟台少监，以工书知名。友贞弱冠时，母病笃，医言唯啖人肉乃差。友贞独念无可求治，乃割股肉以饴亲，母病寻差。则天闻之，令就其家验问，特加旌表。（《旧唐书·卷一百九十二·列传第一百四十二·隐逸》㉕*）
#   [[:w:武則天|武則天]]時期，杭州臨安縣尉薛震好吃人肉，“有債主及奴詣臨安，于客舍，遂飲之醉。殺而臠之，以水銀和煎，并骨消盡。后又欲食其婦，婦覺而遁。縣令詰得其情，申州，錄事奏，奉敕杖殺之。”（《[[:w:朝野僉載|朝野僉載]]》）
#   武則天時期，“周岭南首陳元光設客，令一袍褲行酒。光怒，令曳出，遂殺之。須臾爛煮，以食諸客。后呈其二手，客懼，攫喉而吐。”（出《摭言》。明抄本作出《朝野僉載》）
#   697年: 丁卯，（李）昭德、（来）俊臣同弃市，时人无不痛昭德而快俊臣。仇家争啖俊臣之肉，斯须而尽，抉眼剥面，披腹出心，腾蹋成泥。（《资治通鉴》卷206）
#   张鷟《[[s:朝野僉載_(四庫全書本)/卷2|朝野佥载]]》卷二：“后诛易之昌宗等，百姓脔割其肉，肥白如猪肪，煎炙而食。”
#   唐玄宗開元中葉人[[:w:陳藏器|陳藏器]]（713年－741年）《[[:w:本草拾遺|本草拾遺]]》寫吃人肉可以治病。
#   739年: 内给事牛仙童使幽州，受张守珪厚赂。玄宗怒，命思勖杀之。思勖缚架之数日，乃探取其心，截去手足，割肉而啖之，其残酷如此。（《旧唐书·卷一百八十四·列传第一百三十四·宦官》㉕*）&lt;p&gt; 内给事牛仙童纳张守珪赂，诏付思勖杀之。思勖缚于格，箠惨不可胜，乃探心，截手足，剔肉以食，肉尽乃得死。（《新唐书·卷二百七·列传第一百三十二·宦者上》㉕）&lt;/p&gt;&lt;p&gt;739年: 上（唐玄宗李隆基）怒，甲戌，命杨思勖杖杀之（牛仙童）。思勖缚格，杖之数百，刳取其心，割其肉啖之。（《资治通鉴》卷214）&lt;/p&gt;
#   757年: （鲁）炅城中食尽，煮牛皮筋角而食之，米斗至四五十千，有价无米，鼠一头至四百文，饿死者相枕藉。……炅在围中一年，救兵不至，昼夜苦战，人相食。（《旧唐书·卷一百一十四·列传第六十四·鲁炅等》㉕*）&lt;p&gt;（鲁）炅被围凡一年，昼夜战，人至相食，卒无救。（《新唐书·卷一百四十七·列传第七十二·三王鲁辛冯三李曲二卢》㉕）&lt;/p&gt;
#   757年: 尹子奇攻围（睢阳）既久，城中粮尽，易子而食，析骸而爨，人心危恐，虑将有变。（张）巡乃出其妾，对三军杀之，以飨军士。曰：“诸公为国家戮力守城，一心无二，经年乏食，忠义不衰。巡不能自割肌肤，以啖将士，岂可惜此妇，坐视危迫。”将士皆泣下，不忍食，巡强令食之。乃括城中妇人；既尽，以男夫老小继之，所食人口二三万，人心终不离变。（《旧唐书·卷一百八十七下·列传第一百三十七·忠义下》㉕*）&lt;p&gt;（张）巡士多饿死，存者皆痍伤气乏。巡出爱妾曰：“诸君经年乏食，而忠义不少衰，吾恨不割肌以啖众，宁惜一妾而坐视士饥？”乃杀以大飨，坐者皆泣。巡强令食之，远亦杀奴僮以哺卒，至罗雀掘鼠，煮铠弩以食。……被围久，初杀马食，既尽，而及妇人老弱凡食三万口。人知将死，而莫有畔者。城破，遣民止四百而已。 （《新唐书·卷一百九十二·列传第一百一十七·忠义中》㉕） &lt;/p&gt;&lt;/p&gt;（张巡守睢阳，）茶纸既尽，遂食马；马尽，罗雀掘鼠；雀鼠又尽，巡出爱妾，杀以食士，远亦杀其奴；然后括城中妇人食之；既尽，继以男子老弱。人知必死，莫有叛者，所馀才四百人。 （《资治通鉴》卷220）&lt;/p&gt;
#   758年: 明年，改乾元元年，伪德州刺史王暕、贝州刺史宇文宽等皆归顺，河北诸军各以城守累月，贼使蔡希德、安太清急击，复陷于贼，虏之以归，脔食其肉。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕*）
#   759年: 二年正月，史思明自率范阳精卒复陷魏州，乃伪称燕王。王师虽众，军无统帅，进退无所承禀，自冬徂春，竟未破贼，但引漳水以灌其城，城中食尽，易子而食。（《旧唐书·卷一百二十·列传第七十·郭子仪等》㉕*）&lt;p&gt; （安）庆绪自十月被围至二月，城中人相食，米斗钱七万余，鼠一头直数千，马食隤墙麦鞬及马粪濯而饲之。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕）&lt;/p&gt;&lt;p&gt;（郭子仪军）连营进围相州，引漳水灌城，漫二时，不能破。城中粮尽，人相食。庆绪求救于史思明。（《新唐书·卷一百三十七·列传第六十二·郭子仪》㉕）&lt;/p&gt;&lt;p&gt; 乾元元年秋九月，帝诏郭子仪率九节度兵凡二十万讨庆绪，攻卫州，……王师围已固，筑浚城隍三周，决安阳水灌城。城中栈而处，粮尽，易口以食，米斗钱七万余，一鼠钱数千，屑松饲马，隤墙取麦秸，濯粪取刍，城中欲降不得。（《新唐书·卷二百二十五上·列传第一百五十上·逆臣上》㉕）&lt;/p&gt;
#   760年: 有纳赂于上言求官者，（吕）諲补之蓝田尉。五月，上言事泄笞死，以其肉令从官食之，諲坐贬太子宾客。（《旧唐书·卷一百八十五下·列传第一百三十五·良吏下》㉕*）
#   760年:   三品钱行浸久，属岁荒，米斗至七千钱，人相食。 （《资治通鉴》卷221）
#   760年: 时大雾，自四月雨至闰月末不止。米价翔贵，人相食，饿死者委骸于路。（《旧唐书·卷十·本纪第十·肃宗》㉕*）&lt;p&gt; 是时自四月初大雾大雨，至闰四月末方止。是月，逆贼史思明再陷东都，米价踊贵，斗至八百文，人相食，殍尸蔽地。（《旧唐书·卷三十六·志第十六·天文下》㉕） &lt;/p&gt;&lt;p&gt;乾元三年闰四月，大雾，大雨月余。是月，史思明再陷东都，京师米斗八百文，人相食，殍骸蔽地。（《旧唐书·卷三十七·志第十七·五行》㉕）&lt;/p&gt;
#   761年: 时洛阳四面数百里，人相食，州县为墟。（《旧唐书·卷二百上·列传第一百五十·安禄山等》㉕*）&lt;p&gt; 朝义虚怀礼下，事皆决大臣，然无经略才。当此时，洛阳诸郡人相食，城邑榛墟，（《新唐书·卷二百二十五上·列传第一百五十上·逆臣上》㉕）&lt;/p&gt;
#  [[:w:唐代宗|唐代宗]]廣德元年（763年），江東大疫，“死者過半”，[[:w:獨孤及|獨孤及]]描述這次的災難：“辛丑歲（762年），大旱，三吳飢甚，人相食。明年大疫，死者十七八，城郭邑居為之空虛，而存者無食，亡者無棺殯悲哀之送。大抵雖其父母妻子也啖其肉，而棄其骸於田野，由是道路積骨相支撐枕藉者彌二千里，春秋以來不書。”（《吊道殣文》）&lt;p&gt;江、淮大饥，人相食。（《资治通鉴》卷222）&lt;/p&gt;
#   [[:w:白居易|白居易]]（772年－846年）寫《輕肥》一詩有“是歲江南旱，衢州人食人。”
#   [[:w:張茂昭|張茂昭]]為節鎮，頻吃人肉，及除統軍，到京。班中有人問曰：聞尚書在鎮好人肉，虛實？” 昭笑曰：“人肉腥而且肕，爭堪吃。”（《盧氏雜記》）
#   766年: 监军张志斌自陕入奏，（周）智光馆给礼慢，志斌责其不肃。智光大怒曰：“仆固怀恩岂有反状！皆由尔鼠辈作福作威，惧死不敢入朝。我本不反，今为尔作之。”因叱下斩之，脔其肉以饲从者。（《旧唐书·卷一百一十四·列传第六十四·鲁炅等》㉕*）&lt;p&gt;（周智光）叱下斩之（张志斌），脔食其肉。（《资治通鉴》卷224）&lt;/p&gt;
#   775年:承嗣既令（田）廷玠（或作田庭玠）守沧州，而（李）宝臣、朱滔兵攻击，欲兼其土宇。廷玠婴城固守，连年受敌，兵尽食竭，人易子而食，卒无叛者，卒能保全城守。（《旧唐书·卷一百四十一·列传第九十一·田承嗣等》㉕*）
#   796年: 军士又呼曰：“仓官刘叔何给纳有奸。”杀而食之。（《资治通鉴》卷235）
#   799年: 是日，汴州军乱，杀陆长源及节度判官孟叔度、丘颖，军人脔而食之。（《旧唐书·卷十三·本纪第十三·德宗下》㉕*）&lt;p&gt;兵士怨怒滋甚，乃执长源及叔度等脔而食之，斯须骨肉糜散。（《旧唐书·卷一百四十五·列传第九十五·刘玄佐等》㉕）&lt;/p&gt;&lt;p&gt;才八日，军乱，杀长源及叔度等，食其肉，放兵大掠。（《新唐书·卷一百五十一·列传第七十六·关董袁赵窦》㉕）&lt;/p&gt;&lt;p&gt;是日，军士作乱，杀（陆）长源、（孟）叔度，脔食之，立尽。（《资治通鉴》卷235）&lt;/p&gt;
#   803年: 盐夏节度判官崔文先权知盐州，为政苛刻。冬，闰十月，庚戌，部将李庭俊作乱，杀而脔食之。（《资治通鉴》卷236）
#   807年: 锜不自安，亦请入朝，乃拜锜左仆射。锜乃署判官王澹为留后。既而迁延发期，澹与中使频喻之，不悦，遂讽将士以给冬衣日杀澹而食之。监军使闻乱，遣衙将赵锜慰喻，又脔食之。（《旧唐书·卷一百一十二·列传第六十二·李暠等》㉕*）&lt;p&gt;会使者召锜，称疾，留后王澹为具行，锜怒，阴教士脔食之，即胁使者为众奏天子，幸得留。（《新唐书·卷一百八十一·列传第一百六·陈夷行等》㉕）&lt;/p&gt;&lt;p&gt;807: （李）锜严兵坐幄中，（王）澹与敕使入谒，有军士数百噪于庭曰：“王澹何人，擅主军务！”曳下，脔食之；大将赵琦出慰止，又脔食之（《资治通鉴》卷237）&lt;/p&gt;
#   817年: 蔡将有李端者，过溵河降重胤。其妻为贼束缚于树，脔食至死，将绝，犹呼其夫曰：“善事乌仆射。”（《旧唐书·卷一百六十一·列传第一百一十一·李光进等》㉕*）&lt;p&gt;李湍妻。湍，吴元济之军人也。元和中，淮南未平，湍心怀向顺，乃急渡溵河，东降乌重胤。其妻遂为贼束缚在树，脔而食之，至死，叫其夫曰：“善事乌仆射。”观者义之。至是，重胤以其事请列史册。十三年，宪宗下诏从之。（《旧唐书·卷一百九十四上·列传第一百四十四上·突厥上》㉕）&lt;/p&gt;&lt;p&gt;李湍妻某氏。湍籍吴元济军，元和中，自拔归鸟重胤，妻为贼缚而脔食之，将死，犹号湍曰：“善事鸟仆射！”观者叹泣。重胤请以其事属史官，诏可。（《新唐书·卷二百五·列传第一百三十·列女》㉕）&lt;/p&gt;
#   822年: （王）播至淮南，属岁旱俭，人相啖食，课最不充，设法掊敛，比屋嗟怨。（《旧唐书·卷一百六十四·列传第一百一十四·王播等》㉕*）&lt;p&gt; 是时，南方旱歉，人相食，（王）播掊敛不少衰，民皆怨之。（《新唐书·卷一百六十七·列传第九十二·白裴崔韦二李皇甫王》㉕）&lt;/p&gt;
#   829年: 属岁旱俭，人至相食，楚均富赡贫，而无流亡者。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）
#   832年:（李）听先遣亲吏至徐州慰劳将士，苍头不欲听复来，说军士杀其亲吏，脔食之。（《资治通鉴》卷244）
#   约841年: （杜牧）作《罪言》。其辞曰：……. 山东叛且三五世，后生所见言语举止，无非叛也，以为事理正当如此，沉酣入骨髓，无以为非者，至有围急食尽，啖尸以战。以此为俗，岂可与决一胜一负哉？（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕*）
#   868年: 其年冬，庞勋杀崔彦曾，据徐州，聚众六七万。徐无兵食，乃分遣贼帅攻剽淮南诸郡，滁、和、楚、寿继陷。谷食既尽，淮南之民多为贼所啖。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）&lt;p&gt; 勋还，果盗徐州，其众六七万。徐乏食，分兵攻滁、和、楚、寿，陷之，粮尽，啖人以饱。（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕）&lt;/p&gt;
#   868年: 一日，贼军乘间，步骑径入湘垒，淮卒五千人皆被生絷送徐州，为贼蒸而食之。（《旧唐书·卷一百七十二·列传第一百二十二·令狐楚等》㉕*）&lt;/p&gt;&lt;p&gt;湘乃彻警释械，日与勋众欢言。后贼乘间直袭湘垒，悉俘而食之，醢湘及监军郗厚本。（《新唐书·卷一百六十六·列传第九十一·贾杜令狐》㉕）&lt;/p&gt;
#   868年: 庞勋又令将刘贽攻濠州，陷之，囚刺史卢望回于回车馆，望回郁愤而死，仆妾数人皆为贼蒸而食之。（《旧唐书·卷十九上·本纪第十九上·懿宗》㉕*）
#   869年: 吴迥守濠州，粮尽食人，驱女孺运薪塞隍，并填之，整旅而行，马士举斩以献。（《新唐书·卷一百四十八·列传第七十三·令狐张康李刘田王牛史》㉕*）&lt;p&gt;马举攻濠州，自夏及冬不克，城中粮尽，杀人而食之（《资治通鉴》卷251）&lt;/p&gt;
#   876年：李廷节妻崔。乾符中，廷节为郏城尉。王仙芝攻汝州，廷节被执。贼见崔妹美，将妻之，诟曰：“我，士人妻，死亡有命，奈何受贼污？”贼怒，刳其心食之。（《新唐书·卷二百五·列传第一百三十·列女》㉕*）
#   878年: （李）尽忠械文楚等五人送斗鸡台下，（李）克用令军士玼食之，以骑践其骸。（《资治通鉴》卷253）
#   881年，[[:w:唐僖宗|唐僖宗]]廣明二年：（[[:w:黃巢|黃巢]]攻佔長安，）時京畿百姓皆寨于山谷，累年費耕耘，賊坐空城，賦輸無如，谷食騰踴，米斗三十錢，官軍皆執山寨百姓，蠰于賊為食，人獲數十萬”（《[[:w:舊唐書|舊唐書]]·卷二百下·列传第一百五十·朱泚 黄巢 秦宗权》㉕*）&lt;p&gt; 二年春正月甲辰朔，天下勤王之师，云会京畿，京师食尽。贼食树皮，以金玉买人于行营之师，人获数百万。山谷避乱百姓，多为诸军之所执卖。（《旧唐书·卷十九下·本纪第十九下 僖宗》㉕）&lt;/p&gt;&lt;p&gt;于时畿民栅山谷自保，不得耕，米斗钱三十千，屑树皮以食，有执栅民鬻贼以为粮，人获数十万钱。（《新唐书·卷二百二十五下·列传第一百五十下·逆臣下》㉕）&lt;/p&gt;&lt;p&gt;民避乱皆入深山筑栅自保，农事俱废，长安城中斗米直三十缗。贼（黄巢）卖人于官军以为粮，官军或执山栅之民鬻之，人直数百缗，以肥瘠论价。（《资治通鉴》卷254）&lt;/p&gt;
#  883年，唐僖宗中和三年883年：时黄巢与宗权合从，纵兵四掠，远近皆罹其酷。时仍岁大饥，民无积聚，贼俘人为食，其炮炙处谓之“舂磨寨”，白骨山积，丧乱之极，无甚于斯。（《旧唐书·卷十九下·本纪第十九下 僖宗》㉕*）&lt;p&gt;贼（黄巢）围陈郡百日，关东仍岁无耕稼，人饿倚墙壁间，贼俘人而食，日杀数千。贼有舂磨砦，为巨碓数百，生纳人于臼碎之，合骨而食，其流毒若是。（《旧唐书·卷二百下·列传第一百五十 朱泚 黄巢 秦宗权》㉕）&lt;/p&gt;&lt;p&gt;巢已东，使孟楷攻蔡州。节度使秦宗权迎战，大败，即臣贼，与连和。楷击陈州，败死，巢自围之，略邓、许、孟、洛，东入徐、兖数十州。人大饥，倚死墙堑，贼俘以食，日数千人，乃办列百巨碓，糜骨皮于臼，并啖之。（《新唐书·卷二百二十五下·列传第一百五十下 逆臣下》㉕）&lt;/p&gt;&lt;p&gt;是时，陈州四面，贼寨相望，驱掳编氓，杀以充食，号为“舂磨寨”。（《旧五代史·卷一（梁书）·太祖纪一》㉕）&lt;/p&gt;&lt;p&gt;秦宗权以蔡州附巢，巢势甚盛，乃悉众围犨，置舂磨，糜人之肉以为食。（《新五代史·卷四十二·杂传第三十·朱宣等》㉕）&lt;/p&gt;&lt;p&gt;时民间无积聚，贼（黄巢）掠人为粮，生投于碓硙，并骨食之，号给粮之处曰“舂磨寨”。纵兵四掠，自河南、许、汝、唐、邓、孟、郑、汴、曹、濮、徐、兖等数十州，咸被其毒。 （《资治通鉴》卷255）&lt;/p&gt;
#   884年: （秦宗权）所至屠翦焚荡，殆无孑遗。其残暴又甚于巢，军行未始转粮，车载盐尸以从。北至卫、滑，西及关辅，东尽青、齐，南出江、淮，州镇存者仅保一城，极目千里，无复烟火。（《资治通鉴》卷256）&lt;p&gt; 巢贼虽平，而宗权之凶徒大集，西至金、商、陕、虢，南极荆、襄，东过淮甸，北侵徐、兖、汴、郑，幅员数十州。五六年间，民无耕织，千室之邑，不存一二，岁既凶荒，皆脍人而食，丧乱之酷，未之前闻。（《旧唐书·卷二十上·本纪第二十上·昭宗》㉕*）&lt;/p&gt;&lt;p&gt;（秦宗权）贼首皆慓锐惨毒，所至屠残人物，燔烧郡邑。西至关内，东极青、齐，南出江淮，北至卫滑，鱼烂鸟散，人烟断绝，荆榛蔽野。贼既乏食，啖人为储，军士四出，则盐尸而从。（《旧唐书·卷二百下·列传第一百五十·朱泚 黄巢 秦宗权》㉕）&lt;/p&gt;&lt;p&gt; 中和二年，关内大饥。四年，关内大饥，人相食。（《新唐书·卷三十五·志第二十五 稼穑不成》㉕）&lt;/p&gt;&lt;p&gt;中和四年，江南大旱，饥，人相食。（《新唐书·卷三十五·志第二十五·常旸》㉕）&lt;/p&gt;
#   886年: 荆南、襄阳仍岁蝗旱，米斗三十千，人多相食。（《旧唐书·卷十九下·本纪第十九下·僖宗》㉕*）&lt;p&gt; 光启二年二月，荆、襄大饥，米斗三千钱，人相食。（《新唐书·卷三十五·志第二十五·稼穑不成》㉕）&lt;/p&gt;&lt;p&gt;二年，荆、襄蝗、米斗钱三千，人相食；（《新唐书·卷三十六·志第二十六·五行三》㉕）&lt;/p&gt;
#   886年: （张）瑰固垒二岁，樵苏皆尽，米斗钱四十千，计抔而食，号为“通肠”。疫死者，争啖其尸，县首于户以备馔。（《新唐书·卷一百八十六·列传第一百一十一 ·周王邓陈齐赵二杨顾》㉕*）
#   887年: 戊午，秦彦遣毕师铎、秦稠将兵八千出（扬州）城，西击杨行密。稠败死，士卒死者什七八。城中乏食，樵采路绝，宣州军始食之。（《资治通鉴》卷257）&lt;p&gt;五月，寿州刺史杨行密率兵攻（秦）彦，……重围半年，（扬州）城中刍粮并尽，草根木实、市肆药物、皮囊革带，食之亦尽。外军掠人而卖，人五十千。死者十六七，纵存者鬼形鸟面，气息奄然。（《旧唐书·卷一百八十二·列传第一百三十二·王重荣等》㉕*）&lt;/p&gt;&lt;p&gt;杨行密围扬州，毕师铎厚赍宝币，啖（杜）雄连和。雄率军浮海屯东塘。是时扬州围久，皮囊革带食无余，军中杀人代粮，才千钱。（《新唐书·卷一百九十·列传第一百一十五·三刘成杜钟张王》㉕）&lt;/p&gt;&lt;p&gt;是时，城中仓廪空虚，饥民相杀而食，其夫妇、父子自相牵，就屠卖之，屠者刲剔如羊豕。（《新五代史·卷六十一·吴世家第一》㉕）&lt;/p&gt;
#   887年: （高）骈家属并在道院，秦彦供给甚薄，薪蒸亦阙。奴仆彻延和阁栏槛煮革带食之，互相篡啖。（《旧唐书·卷一百八十二·列传第一百三十二·王重荣等》㉕*）&lt;p&gt;高骈在道院，秦彦供给甚薄，左右无食，至然木像、煮革带食之，有相啖者。（《资治通鉴》卷257）&lt;/p&gt;
#   887年，光启三年：（杨）行密攻围（广陵）弥急，城中食尽，米斗四十千，居人相啖略尽。十月，城陷，秦、毕走东塘，行密入广陵，辇外寨之粟以食饥民，即日米价减至三千。（《旧五代史·卷一百三十四·僭伪列传一》㉕*）&lt;p&gt;[[:w:杨行密|杨行密]]围广陵且半年，秦彦、毕师铎大小数十战多不利，城中无食，料值钱五十缗，草根木实皆尽，以堇泥为饼食之，饿死者大半。宣州军掠人诣肆卖之，驱缚屠割如羊豕，讫无一声，流血满于坊市。彦、师铎无如之何，颦蹙而已。（《资治通鉴》卷257）&lt;/p&gt;
#   887年: 周迪妻某氏。迪善贾，往来广陵。会毕师铎乱，人相掠卖以食。迪饥将绝，妻曰：“今欲归，不两全。君亲在，不可并死，愿见卖以济君行。”迪不忍，妻固与诣肆，售得数千钱以奉。迪至城门，守者谁何，疑其绐，与迪至肆问状，见妻首已在枅矣。迪里余体归葬之。（《新唐书·卷二百五·列传第一百三十·列女》㉕*）
#   888年: （李）罕之与（张）言甚笃，然性猜暴。是时大乱后，野无遗秆，部卒日剽人以食。《新唐书·卷一百八十七·列传第一百一十二·二王诸葛李孟》㉕*）&lt;p&gt;时大乱之后，野无耕稼，罕之部下以俘剽为资，啖人作食。……自是罕之日以兵寇钞怀、孟、晋、绛，数百里内，郡邑无长吏，闾里无居民。……自是数州之民，屠啖殆尽，荆棘蔽野，烟火断绝，凡十余年。（《旧五代史·卷十五（梁书）·列传五》㉕）&lt;/p&gt;&lt;p&gt;罕之留其子颀事晋，乃之泽州，日以兵钞怀、孟间，啖人为食。（《新五代史·卷四十二·杂传第三十·朱宣等》㉕）&lt;/p&gt;&lt;p&gt;（李）罕之勇而无谋，性复贪暴，意轻（张）全义，闻其勤俭力穑，笑曰：“此田舍一夫耳！”…….（李）罕之所部不耕稼，专以剽掠为资，啖人为粮。……（李罕之）以寇钞为事，自怀、孟、晋、绛数百里间，州无刺史，县无令长，田无麦禾，邑无烟火者，殆将十年。（《资治通鉴》）&lt;/p&gt;
#   889年，[[:w:唐昭宗|唐昭宗]]龍紀元年：楊行密圍宣州，城中食盡，人相啖……（《資治通鑒》卷258）
#   891年: 会吏盗减诸军禀食，（王）建怒其众曰：“招讨吏之谋也。”纵士执之，醢食于军。（《新唐书·卷二百二十四下·列传第一百四十九下·叛臣下》㉕*）&lt;p&gt;一日，（王）建阴令军士于行府门外擒（韦）昭度亲吏，脔而食之，（王）建徐启（韦）昭度曰：“盖军士乏食，以至于是耶！”昭度大惧，遂留符节与建，即日东还。（《旧五代史·卷一百三十六·僭伪列传三》㉕）&lt;/p&gt;&lt;p&gt;昭度迟疑未决，建遣军士擒昭度亲吏于军门，脔而食之，建入白曰：“军士饥，须此为食尔！”昭度大恐，即留符节与建而东。（《新五代史·卷六十三·前蜀世家第三》㉕）&lt;/p&gt;&lt;p&gt;庚子，（王）建阴令东川将唐友通等擒（韦）昭度亲吏骆保于行府门，脔食之，云其盗军粮。（《资治通鉴》卷258）&lt;/p&gt;
#   891年: 孙儒悉焚扬州庐舍，尽驱丁壮及妇女渡江，杀老弱以充食。（《资治通鉴》卷258）
#   893年: 景福二年春，（李克用）大举以伐王镕，……王镕出师三万来援，武皇（李克用）逆战于叱日岭下，镇人败，斩首万余级。时岁饥，军乏食，脯尸肉而食之。（《旧五代史·卷二十六（唐书）·武皇纪下》㉕*）&lt;p&gt;（李克用的）河东军无食。脯其尸而啖之。 （《资治通鉴》卷259）&lt;/p&gt;
#   894年: 王建攻彭州，城中人相食（《资治通鉴》卷259）
#   902年，唐昭宗天复二年：是冬，大雪，（凤翔）城中食尽，冻馁死者不可胜计，或卧未死，肉已为人所。市中卖人肉斤直钱百，犬肉值五百。”（《资治通鉴》卷263）&lt;p&gt;昭宗在凤翔，为梁兵所围，城中人相食，父食其子，而天子食粥，六宫及宗室多饿死。其穷至于如此，遂以亡。（《新唐书·卷五十二·志第四十二·食货二》㉕*）&lt;/p&gt;&lt;p&gt;（朱温的后）梁军围之（凤翔）逾年，（李）茂贞每战辄败，闭壁不敢出。城中薪食俱尽，自冬涉春，雨雪不止，民冻饿死者日以千数。米斗直钱七千，至烧人屎煮尸而食。父自食其子，人有争其肉者，曰：“此吾子也，汝安得而食之！”人肉斤直钱百，狗肉斤直钱五百。父甘食其子，而人肉贱于狗。天子于宫中设小磨，遣宫人自屑豆麦以供御，自后宫、诸王十六宅，冻馁而死者日三四。城中人相与邀遮茂贞，求路以为生。（《新五代史·卷四十·杂传第二十八·李茂贞等》㉕）&lt;/p&gt;

==五代十國==
#   906年:天祐三年，（朱）全忠自将攻沧州，……全忠环沧筑而沟之，内外援绝，人相食。（刘）仁恭求战，不许。（《新唐书· 卷二百一十二·列传第一百三十七·藩镇卢龙》㉕*）&lt;p&gt;汴人深沟高垒以攻沧州，内外阻绝，（刘）仁恭不能合战，城中大饥，人相篡啖，析骸而爨，丸土而食，转死骨立者十之六七。……城中乏食，米斗直三万，人首一级亦直十千，军士食人，百姓食墐土，驴马相遇，食其鬃尾，士人出入，多为强者屠杀。（《旧五代史·卷一百三十五·僭伪列传二》㉕）&lt;/p&gt;&lt;p&gt;梁军壁长芦，深沟高垒，（刘）仁恭不能近。沧州被围百余日，城中食尽，人自相食，析骸而爨，或丸墐土而食，死者十六七。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕）&lt;/p&gt;&lt;p&gt;时汴军筑垒围沧州，鸟鼠不能通。（刘）仁恭畏其强，不敢战。城中食尽，丸土而食，或互相掠啖。（《资治通鉴》卷265）&lt;/p&gt;
#   909年:（刘）守文将吏孙鹤、吕兖等，立守文子延祚以距（刘）守光，守光围之百余日，城中食尽，米斛直钱三万，人相杀而食，或食墐土，马相食其骏尾，（吕）兖等率城中饥民食以麹，号“宰务”，日杀以饷军。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕*）&lt;p&gt;刘守光围沧州久不下，执刘守文至城下示之，犹固守。城中食尽，民食堇泥，军士食人，驴马相啖尾。吕兖选男女羸弱者，饲以黮面而烹之，以给军食，谓之宰杀务。 （《资治通鉴》卷267）&lt;/p&gt;
#  911: （刘）守光大怒，推之（孙鹤）伏锧，令军士割其肉生啖之。鹤大呼曰：“百日之外，必有急兵矣！”守光命窒其口，寸斩之，有识为之嗟惋。（《旧五代史·卷一百三十五·僭伪列传二》㉕*）&lt;p&gt;（刘）守光怒，推之（孙鹤）伏锧，令军士割而啖之。（《新五代史·卷三十九·杂传第二十七·王镕等》㉕）&lt;/p&gt;&lt;p&gt;（刘）守光怒，伏诸质上，令军士剐而啖之。鹤呼曰：“百日之外，必有急兵！”守光命以土窒其口，寸斩之。（《资治通鉴》卷268）&lt;/p&gt;
#  916: 晋人围贝州逾年，城中食尽，啖人为粮。（《资治通鉴》卷269）
#   922年: （李存勖）获（张）处球、处瑾、处琪并其母，及同恶高濛李翥、齐俭等，皆折足送行台，镇人请醢而食之；（《旧五代史·卷二十九（唐书）·庄宗纪三》㉕*）
#  925年，後唐莊宗同光三年： （郭）崇韬欲诛（王）宗弼以自明，己巳，白（李）继岌收宗弼及王宗勋、王宗渥，皆数其不忠之罪，族诛之，籍没其家。蜀人争食宗弼之肉。 （《资治通鉴》卷274）
#   929年: （董璋）遣其将李彦钊扼剑门关为七砦，于关北增置关，号永定。凡唐戍兵东归者，皆遮留之，获其逃者，覆以铁笼，火炙之，或刲肉钉面，割心而啖。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕*）
#  930: （董）璋怒，令军士十人，持刀刲割其（姚洪）肤，燃镬于前，自取啖食，洪至死大骂不已。（《旧五代史·卷七十（唐书）·列传二十二》㉕*）&lt;p&gt;（董）璋怒，然镬于前，令壮士十人刲其肉而食，洪至死大骂。（《新五代史·卷三十三·死事传第二十》㉕）&lt;/p&gt;&lt;p&gt;（董）璋怒，然镬于前，令壮士十人刲其（姚洪）肉自啖之，洪至死骂不绝声。（《资治通鉴》卷277）&lt;/p&gt;
#   约930年:（李）赞华好饮人血，姬妾多刺臂以吮之；婢仆小过，或抉目，或刀刲火灼；夏氏不忍其残，奏离婚为尼。 （《资治通鉴》卷277）
#  934: （薛）文杰善数术，自占云：“过三日可无患。”送者闻之，疾驰二日而至，军士踊跃，磔文杰于市，闽人争以瓦石投之，脔食立尽。（《新五代史·卷六十八·闽世家第八》㉕*）&lt;p&gt;（薛）文杰出，（王）继鹏伺之于启圣门外，以笏击之仆地，槛车送军前，市人争持瓦砾击之。文杰善术数，自云过三日则无患。部送者闻之，倍道兼行，二日而至，士卒见之踊跃，脔食之（《资治通鉴》卷278）&lt;/p&gt;
#   约942年: （石）信所至黩货，好行杀戮。军士有犯法者，信召其妻子，对之刲剔支解，使自食其肉，血流盈前，信命乐饮酒自如也。（《新五代史·卷十八·汉家人传第六》㉕*）
#   944年: 同（州）、华（州）奏，人民相食。（《旧五代史·卷七十（唐书）·列传二十二》㉕*）
#   944年: （后晋少帝石重贵）命李守贞、符彦卿率师东讨。（杨）光远素无兵众，惟婴城（青州）自守，守贞以长连城围之。冬十一月，（杨）承勋与弟承信、承祚见城中人民相食将尽，知事不济，劝（杨）光远乞降，冀免于赤族。（《旧五代史·卷九十七（晋书）·列传十二》㉕*）&lt;p&gt;契丹已北，出帝（石重贵）复遣（李守贞、符彦卿东讨，光远婴城固守，自夏至冬，城中人相食几尽。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕）&lt;/p&gt;
#  945年: 闽人或告福州援兵谋叛，闽主（王）延政收其铠仗，遣还，伏兵于隘，尽杀之，死者八千馀人，脯其肉以归为食。 （《资治通鉴》卷284）
#  947年: （杨）承勋事晋为郑州防御使，（耶律）德光灭晋，使人召承勋至京师，责其劫父，脔而食之。（《新五代史·卷五十一·杂传第三十九·朱守殷等》㉕*）&lt;p&gt;戊子，（辽军）执郑州防御使杨承勋至大梁，责以杀父叛契丹，命左右脔食之。（《资治通鉴》卷286）&lt;/p&gt;
#  947年，后晋天福十二年（947年：大同元年春正月……己丑，以张彦泽擅徙重贵开封，杀桑维翰，纵兵大掠，不道，斩于市。晋人脔食之。（《辽史· 卷四·本纪第四·太宗下》㉕*）&lt;p&gt;戎王（辽太宗耶律德光）知其（张彦泽）众怒，遂令弃市，仍令高勋监决，断腕出锁，然后刑之。勋使人剖其心以祭死者，市人争其肉而食之。（《旧五代史·卷九十八（晋书）·列传十三》㉕）&lt;/p&gt;&lt;p&gt;百官皆请不赦（张彦泽），而都人争投状疏其恶，乃命高勋监杀之。彦泽前所杀士大夫子孙，皆缞绖杖哭，随而诟詈，以杖朴之，彦泽俯首无一言。行至北市，断腕出锁，然后用刑，勋剖其心祭死者，市人争破其脑，取其髓，脔其肉而食之。（《新五代史·卷五十二·杂传第四十·杜重威等》㉕）&lt;/p&gt;&lt;p&gt;己丑，斩（张）彦泽、（傅）住皃于北市，仍命高勋监刑。彦泽前所杀士大夫子孙，皆绖杖号哭，随而诟詈，以杖扑之。勋命断腕出锁，剖其心以祭死者。市人争破其脑取髓，脔其肉而食之。 （《资治通鉴》卷286）&lt;/p&gt;
#  948年: （苏）逢吉等秘不发丧，下诏称：“（杜）重威父子，因朕小疾，谤议摇众，皆斩之。”磔死于市，市人争啖其肉。（《旧五代史·卷一百（汉书）·高祖纪下》㉕*）&lt;p&gt;磔（杜）重威尸于市，市人争啖其肉，吏不能禁，斯须而尽。 （《资治通鉴》卷287）&lt;/p&gt;
#   948年: （李）守贞自谓天时人事合符于己，乃潜结草贼，令所在窃发，遣兵据潼关。朝廷命白文珂、常思等领兵问罪，复遣枢密使郭威西征。……既而城中粮尽，杀人为食。（《旧五代史·卷一百九（汉书）·列传六》㉕*）&lt;p&gt;（李）守贞（潼关）城中兵无几，而食又尽，杀人而食。（《新五代史·卷五十二·杂传第四十·杜重威等》㉕）&lt;/p&gt;
#  949年，後漢高祖乾佑元年二年：（赵）思绾粮尽，城中人相食（宋）（《宋史· 卷二百五十二·列传第十一·王景等》㉕*）&lt;p&gt;朝廷闻之，命郭从义、王峻帅师伐之（赵思绾）。及攻其城（长安），王师伤者甚众，乃以长堑围之。经年粮尽，遂杀人充食。思绾尝对众取人胆以酒吞之，告众曰：“吞此至一千，即胆气无敌矣。”（《太平广记》：贼臣赵思绾自倡乱至败，凡食人肝六十六，无不面剖而脍之。）（《旧五代史·卷一百九（汉书）·列传六》㉕）&lt;/p&gt;&lt;p&gt;隐帝（后汉隐帝刘承祐）遣郭威西督诸将兵，先围（李）守贞于河中。居数月，（赵）思绾城中食尽，杀人而食，每犒宴，杀人数百，庖宰一如羊豕。思绾取其胆以酒吞之，语其下曰：“食胆至千，则勇无敌矣！” （《新五代史·卷五十三·杂传第四十一·王景崇等》㉕）&lt;/p&gt;&lt;p&gt;赵思绾好食人肝，常面剖而脍之，脍尽，人犹未死。又好以酒吞人胆，谓人曰：吞此千数，则胆无敌矣。长安城中食尽，取妇女幼稚为军粮，日计数而给之。每犒军，辄屠数百人，如羊豖法。（《资治通鉴》卷288）&lt;/p&gt;
#  950年: （马希萼）脔食李弘皋、（李）弘节、唐昭胤、杨涤。（《资治通鉴》）
#   苌从简（后唐、后晋武将），陈州人也。……好食人肉，所至多潜捕民间小儿以食。（《新五代史·卷四十七·杂传第三十五·华温琪等》㉕*）
#  [[:w:吴国 (五代十国)|吳國]]將領[[:w:高澧|高澧]]「嗜殺人而飲血，日暮，必於宅前，後掠行人而食之」。（《南村辍耕录》引《九国志》）

==辽宋金==
从《宋史》开始，二十五史开始频繁记载割肉疗亲的尽孝的故事，这反映了儒家伦理和人肉治病理念的普及，宋朝官方是褒奖这种做法的，之后元朝法律禁止，明清官方态度有所保留，但屡禁不止，愈演愈烈。

* 冠冕百行莫大于孝，范防百为莫大于义。先王兴孝以教民厚，民用不薄；兴义以教民睦，民用不争。率天下而由孝义，非履信思顺之世乎。太祖、太宗以来，子有复父仇而杀人者，壮而释之；刲股割肝，咸见褒赏；至于数世同居，辄复其家。一百余年，孝义所感，醴泉、甘露、芝草、异木之瑞，史不绝书，宋之教化有足观者矣。作《孝义传》。《宋史· 卷四百五十六·列传第二百一十五·孝义》

岳飞《满江红》的“壮志饥餐胡虏肉，笑谈渴饮匈奴血”可能是大众文化中最广泛流传的称赞吃人的文学作品。

#  辽穆宗时期（951年-969年）：初，女巫肖古上延年药方，当用男子胆和之。不数年，杀人甚多，至是（957年，应历七年），觉其妄，辛巳，射杀之。（《辽史·卷六·本纪第六·穆宗上》㉕*）&lt;p&gt;京师置百尺牢以处系囚。盖其（辽穆宗）即位未久，惑女巫肖古之言，取人胆合延年药，故杀人颇众。后悟其诈，以鸣镝丛射、骑践杀之。（《辽史·卷六十一·志第三十·刑法志上》㉕）&lt;/p&gt;
#   963年: 众皆感愤，遂破其众于平津亭，擒（张）文表脔而食之。（《宋史· 卷四百八十三·列传第二百四十二·世家六》㉕*）
#  963年乾德元年：（李）处耘释所俘体肥者数十人，令左右分啖之，黥其少健者，令先入朗州。 （《宋史· 卷二百五十七·列传第十六· 吴廷祚等》㉕*）
#  969年，開寶二年（969）：[[:w:王彥昇|王彥昇]]改防州防御使，是冬，又移原州（甘肅鎮原）。 西人（甘肅少數民族）有犯漢法者，彥升不加刑，召僚屬飲宴，引所犯，以手捽斷其耳，大嚼，巵酒下之。其人流血被體，股栗不敢動。前後啗者數百人。西人畏之，不敢犯塞。（[[:w:王辟之|王辟之]]《澠水燕談錄》，《宋史·卷二百五十·列传第九·王彥昇》㉕*）
#  970年，开宝三年：命分司西京。（王）继勋残暴愈甚，强市民家子女备给使，小不如意，即杀食之，而棺其骨弃野外。……长寿寺僧广惠常与继勋同食人肉，令折其胫而斩之。洛民称快。（《宋史· 卷四百六十三·列传第二百二十二·外戚上》㉕*）
#  1006年: 三年，（德恭）被疾，子承庆刲股肉食之。（《宋史· 卷二百四十四·列传第三·宗室一》㉕*）
#  1048年，[[:w:宋仁宗|宋仁宗]]庆历八年：明年，河北大饥，人相食，（子）鼎经营赈救，颇尽力。（《宋史·卷三百·列传第五十九·杨偕等》㉕*）&lt;p&gt;河北、京東西大水為災，人相食，流民入京東者不可勝數（《[[:w:續資治通鑑|續資治通鑑]]》卷50）&lt;/p&gt;
#  约1053年，宋仁宗时期：[[:w:侬智高|（侬）智高]]母[[:w:阿侬|阿侬]]有计谋，智高攻陷城邑，多用其策，僭号皇太后，性惨毒，嗜小儿肉，每食必杀小儿。（《宋史· 卷四百九十五·列传第二百五十四·蛮夷三》㉕*） 
#  1087年，[[:w:宋哲宗|宋哲宗]]元祐二年，[[:w:苏辙|苏辙]]《因旱乞许群臣面对言事剳子》：“臣伏见二年以来，民气未和，天意未顺，災沴荐至，非水即旱。淮南饥饉，人至相食。河北流移，道路不绝。京东困弊，盗贼群起。二圣遇災忧惧，顷发仓廪以救其乏绝，独此三路所散，已仅三百万斛矣！異时赈賉未见此比。然而民力已困，国用己竭，而旱势未止，夏麦失望，秋稼未立，数月之后，公私无继，群盗蜂起，势有必至，臣未知朝廷何以待此？……”
#  1102年: （高永年）行三十里，逢羌帐下亲兵，皆永年昔所推纳熟户也。永年不之备，羌遽执永年以叛，遂为多罗巴所杀，探其心肝食之，谓其下曰：“此人夺我国，使吾宗族漂落无处所，不可不杀也。”（《宋史· 卷四百五十三·列传第二百一十二·忠义八》㉕*）
#  1118年，辽天庆八年（宋重和元年，1118年），十二月，“宁昌军（治懿州）节度使刘宏（无可考）以懿州（治宁昌，今阜新市东北之塔营子村）户三千降金。时山前诸路（此指辽东，非燕山之南）大饥，乾（辽宁北镇南）显（北镇北）宜（义县）锦（锦州市）兴中（朝阳市）等路，斗粟值数缣，民削榆皮食之，既而人相食。”（《辽史· 卷二十八·本纪第二十八·天祚皇帝二》㉕*）
#  1121年: 贼（霍成富）怒，脔其（詹良臣）肉，使自啖之。良臣吐且骂，至死不绝声，见者掩面流涕，时年七十二。（《宋史· 卷四百四十六·列传第二百五·忠义一》㉕*）
#  “甲辰宣和六年（1124年）时转粮给燕山（府治北京西南）民力疲困，重以盐额科敛，加之连年凶荒，民食榆皮野菜不给，至自相食。于是饥民并起为盗。山东有张万仙者，众十万，号敢炽。张迪者，众五万，围濬州（濬州，平川军，治滑州黎阳）五日而去。濬州去京纔一百六十里，而初不知。河北有高托山者，号三十万。其余一二万者，不可胜计也。”（《九朝编年备要卷二十九》）
#  [[:w:宋徽宗|宋徽宗]]宣和七年（1125年）十二月，金两路攻宋。王禀皆破之，“然人众乏粮，三军先食牛马骡，次烹弓弩皮甲，百姓煮萍实、糠籺、草茭以充腹，既而人相食。[九月]城破，禀犹率羸卒巷战，突围出，金兵追之急，遂负太原庙中太宗御容赴汾水死，子荀殉之。”（《续资治通鉴卷九十七》）
#  1125年: 刘敏行，平州人。登天会三年进士。除太子校书郎，累迁肥乡令。岁大饥，盗贼掠人为食。诸县老弱入保郡城，不敢耕种，农事废，畎亩荒芜。（《金史· 卷一百二十八·列传第六十六·循吏》㉕*）
#  1129年：（建炎）三年，山东郡国大饥，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1131年: 有孙知微者，以朝请大夫通判舒州。绍兴元年，贼刘忠入其境，执知微以去，知微不屈，忠怒，脔而食之。（《宋史· 卷四百五十三·列传第二百一十二·忠义八》㉕*）
#  1131年：五湖捕鱼人夏宁聚众千余，掠人为食，郭仲威余党出没淮南，邵青据通州，光世皆招降之。（《宋史·卷三百六十九·列传第一百二十八·张俊》㉕*）&lt;p&gt;五湖捕魚人夏寧，“聚其徒為盜，後有眾千餘，專掠人以為食，……寧等無食，半月之間復啖萬餘人，是日，始具舟迎之。由是江北鄉村愈覺凋殘矣。”（《续资治通鉴卷一零九》）&lt;/p&gt;
#  约1133年，宋高宗紹興三年：唐初，贼朱粲以人为粮，置捣磨寨，谓“啖醉人如食糟豚”。每览前史，为之伤叹。而自靖康丙午岁，金人乱华，六七年间，山东、京西、淮南等路，荆榛千里，斗米至数十千，且不可得。盗贼、官兵以至居民，更互相食。人肉之价，贱于犬豕，肥壮者一枚不过十五千，全躯暴以为腊。登州范温率忠义之人，绍兴癸丑岁泛海到钱唐，有持至行在犹食者。老瘦男子 词谓之“饶把火”，妇人少艾者名为“不羡羊”，小儿呼为“和骨烂”，又通目为“两脚羊”。唐止朱粲一军，今百倍于前世，杀戮焚溺饥饿疾疫陪堕，其死已众，又加之以相食。杜少陵谓“丧乱死多门”，信矣！不意老眼亲见此时，呜呼痛哉！ （莊綽《雞肋編》）
#  [[:w:宋宁宗|宋宁宗]]嘉定年間，[[:w:林千之|林千之]]任西欽州知州，得了一种病（末疾），有個醫士告訴他，吃童女的肉可以強筋健骨。于是，林千之派人在本州境內捕少女，制成肉乾，叫做“地雞”。&lt;ref&gt;王永寬《中國古代酷刑》&lt;/ref&gt;
#  1210年:（嘉定）三年春，建康府大飢，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1215年: 此數人者（李全等造反者），出沒島崓，寶貨山委而不得食，相率食人。（《宋史· 卷四百七十六·列傳第二百三十五·叛臣中》㉕*）
#  1215年: 乙亥，中都降。（王）檝进言曰：“国家以仁义取天下，不可失信于民，宜禁虏掠，以慰民望。”时城中绝粒，人相食，乃许军士给粮，入城转粜，故士得金帛，而民获粒食。（《元史· 卷一百五十三·列传第四十·刘敏等》㉕*）
#  1216: 是春，河朔人相食。（《金史· 卷二十三·志第四·五行》㉕*）&lt;p&gt;四年，河北行省侯摯言：“河北人相食，觀、滄等州鬥米銀十餘兩。（《金史· 卷五十·志第三十一·食貨五》㉕）&lt;/p&gt;&lt;p&gt;金人迁汴，河朔盗起，……太师、国王木华黎兵至城下，……是时兵乱，民废农耕，所在人相食。（《元史· 卷一百五十一·列传第三十八·薛塔剌海等》㉕）&lt;/p&gt;
#  1216年: 邸顺，保定行唐人，岁甲戌，（邸顺）率众来归（元），（元）太祖授行唐令。……丙子，真定饥，群盗据城叛，民皆穴地以避之，盗发地而啖其人，顺擒数百人杀之。（《元史· 卷一百五十一·列传第三十八·薛塔剌海等》㉕*）
#  1224: 十一月……壬子，京城人相食。癸醜，詔曹門、宋門放士民出就食。（《金史· 卷十八·本紀第十八·哀宗下》㉕*）
#  1227年: 时（李）全在围一年，食牛马及人且尽，将自食其军。初军民数十万，至是余数千矣。（《宋史· 卷四百七十七·列传第二百三十六·叛臣下》㉕*）
#  1228年: （完颜）白撒辈纵军四出，剽掠俘虏，挑掘焚炙，靡所不至。哭声相接，尸骸盈野。都尉高禄谦、苗用秀辈仍掠人食之，而白撒诛斩在口，所过官吏残虐不胜，一饭之费有数十金不能给者，公私皇皇，日皆徯大兵至矣。（《金史· 卷一百十三·列传第五十一·完颜赛不等》㉕*）
#  1232年: 时汴京内外不通，米升银二两。百姓粮尽，殍者相望，缙绅士女多行乞于市，至有自食其妻子者，至于诸皮器物皆煮食之，贵家第宅、市楼肆馆皆撤以爨。（《金史· 卷一百十五·列传第五十三·完颜奴申等》㉕*）
#  1233年，绍定六年（1233年）：（南宋大将[[:w:史嵩之|史嵩之]]围唐州，）城中粮尽，人相食，金将乌库哩黑汉，杀其爱妾以啖士，士争杀其妻子（《金史· 卷一百二十三·列传第六十一·忠义三》㉕*，《续资治通鉴·宋纪》）&lt;p&gt;乙酉，大元召宋兵攻唐州，元帅右监军乌古论黑汉死于战，主帅蒲察某为部曲兵所食。城破，宋人求食人者尽戮之，余无所犯。（《金史· 卷十八·本纪第十八·哀宗下》㉕）&lt;/p&gt;
#  1233: 国用安，先名安用，本名咬儿，淄州人。红袄贼杨安儿、李全余党也。……移兵攻徐，（国）用安投水死，求得其尸，剖面系马尾，为怨家田福一军脔食而尽。（《金史· 卷一百十七·列传第五十五·徒单益都等》㉕*）
#  1234年: 端平元年正月辛丑，黑气压（蔡州）城上，日无光，降者言：“城中绝粮已三月，鞍靴败鼓皆糜煮，且听以老弱互食，诸军日以人畜骨和芹泥食之，又往往斩败军全队，拘其肉以食，故欲降者众。”（《宋史· 卷四百一十二·列传第一百七十一·孟珙》㉕*）
#  1234年:甲午，蔡州破，金主自焚死。时汴梁受兵日久，岁饥，人相食，速不台下令纵其民北渡以就食。（《元史· 卷一百二十一·列传第八·速不台》㉕*）
#  约1237: 岁大饥，人相食，留守别之杰讳不诘，（徐）鹿卿命掩捕食人者，尸诸市。（《宋史· 卷四百二十四·列传第一百八十三·陆持之》㉕*）
#  1272年:咸淳七年，江南大饥。八年冬，襄阳饥，人相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）
#  1276: 德祐二年正月，扬州饥。三月，扬州谷价腾踊，民相食。（《宋史· 卷六十七·志第二十·五行五》㉕*）&lt;p&gt;阿术攻扬（州）久不拔，乃筑长围困之。冬，城中食尽，死者满道。明年二月，饥益甚，赴濠水死者日数百，道有死者，众争割啖之立尽。……兵有烹子而食者，犹日出苦战。（《宋史·卷四百二十一·列传第一百八十·杨栋等》㉕）&lt;/p&gt;
#  1277: 十一月，泸州食尽，人相食，遂破之，安抚王世昌自经死。（《宋史· 卷四百五十一·列传第二百一十·忠义六》㉕*）
#  益州双流人周善敏，丧父，庐于墓侧。母病，又割股肉以啖之，遂愈。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  杨庆，鄞人。父病，贫不能召医，乃刲股肉啖之，良已。其后母病不能食，庆取右乳焚之，以灰和药进焉，入口遂差，久之乳复生。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  （伊）审征幼以孝闻，母病，割股肉啖之。（《宋史· 卷四百七十九·列传第二百三十八·世家二》㉕*）
#  刘孝忠，并州太原人。母病经三年，孝忠割股肉、断左乳以食母；（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  吕升，莱州人。父权失明，剖腹探肝以救父疾，父复能视而升不死。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  成象，渠州流江人。以诗书训授里中，事父母以孝闻。母病，割股肉食之，诏赐束帛醪酒。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  庞天祐，江陵人。以经籍教授里中。父疾，天祐割股肉食之；（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  张伯威，大安军人。……大母黄，年九十八，不忍之官。黄得血痢疾濒殆，伯威剔左臂肉食之，遂愈。继母杨因姑病笃，惊而成疾，伯威复剔臂肉作粥以进，其疾亦愈。伯威妹嫁崔均，其姑王疾，妹亦剔左臂肉作粥以进，达旦即愈。（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  母病，（奎）辄割股肉和药以进，母遂愈。（《宋史· 卷三百二十四·列传第八十三·石普》㉕*）
#  （张）掞幼笃孝，蕴病，刲股肉以疗。（《宋史· 卷三百三十三·列传第九十二·杨佐等》㉕*）
#  （常）真妻病，子晏割股肉以养母（《宋史· 卷四百五十六·列传第二百一十五·孝义》㉕*）
#  有朱云孙妻刘氏，姑病，云孙刲股肉作糜以进而愈。姑复病，刘亦刲股以进，又愈。尚书谢谔为赋《孝妇诗》。（《宋史· 卷四百六十·列传第二百一十九·列女》㉕*）
#  聂孝女，字舜英，尚书左右司员外郎天骥之长女也。……崔立劫杀宰相，天骥被创甚，日夜悲泣，恨不即死。舜英谒医救疗百方，至刲其股杂他肉以进，而天骥竟死。时京城围久食尽，……葬其父之明日，绝脰而死。一时士女贤之，有为泣下者。（《金史· 卷一百三十·列传第六十八·列女》㉕*）
#   呼延赞，并州太原人。……其子尝病，赞刲股为羹疗之。（《宋史·卷二百七十九·列传第三十八· 王继忠等》㉕*）
#   蒋偕，字齐贤，华州郑县人。幼贫，有立志。父病，尝刲股以疗，父愈，诘之曰：“此岂孝邪？”曰：“情之所感，实不自知也。”（《宋史·卷三百二十六·列传第八十五·景泰等》㉕*）
#   邑人朱氏女刲股愈母疾，人颂传之，以为治化所致。（《宋史·卷三百四十八·列传第一百七·傅楫等》㉕*）
#   甲幼孤多难，母病，刲股以进。（《宋史·卷三百九十七·列传第一百五十六·徐谊等》㉕*）
#   赵葵，字南仲，京湖制置使方之子。……葵母疾，谒告省侍不得，刲股杂药以寄之。母卒，葵求解官，不许，不得已，卒哭复视事。（《宋史·卷四百一十七·列传第一百七十六·乔行简等》㉕*）
#   陈宗，永嘉人。年十六，母蔡病笃，刲股为饵，病愈。已而复病不救，宗一恸而绝。（《宋史·卷四百五十六·列传第二百一十五·孝义》㉕*）
#   吕仲洙女，名良子，泉州晋江人。父得疾濒殆，女焚香祝天，请以身代，刲股为粥以进。（《宋史·卷四百六十·列传第二百一十九·列女》㉕*）

==元==
元朝法律禁止割肉疗亲，“诸为子行孝，辄以割肝、刲股、埋儿之属为孝者，并禁止之。（《元史· 卷一百五·志第五十三·刑法四》）”但《元史》记载了诸多此般事迹，可见屡禁不止，可能也反映了蒙汉的文化差异。

#   1262年:（中统三年），五月庚申，筑环城（济南）围之；甲戌，围合。（李）鋋自是不得复出，……分军就食民家，发其盖藏以继，不足，则家赋之盐，令以人为食。（《元史·卷二百六·列传第九十三·叛臣》㉕*）
#   1301: 行省右丞刘深远征八百媳妇国，此乃得已而不已之兵也。彼荒裔小邦，远在云南之西南又数千里，……深欺上罔下，帅兵伐之，经过八番，纵横自恣，恃其威力，虐害居民，中途变生，所在皆叛。深既不能制乱，反为乱众所制，军中乏粮，人自相食，（《元史·卷一百六十八·列传第五十五·陈祐（天祥）等》㉕*）
#   1308年:（至大元年六月）河南、山东大饥，有父食其子者，以两道没入赃钞赈之。（《元史· 卷二十二·本纪第二十二·武宗一》㉕*）
#  1319年:延佑六年秋七月丙辰，“来安路总管岑世兴叛，据唐兴州”，杀兼州知州[[:w:黄克仁|黄克仁]]，分食其尸。&lt;ref&gt;《新元史·卷二百四十八·列传第一百四十五》；《招捕总录》&lt;/ref&gt;
#   约1329年: 贼稍引去，（褚不华）乃出，抵杨村桥，贼奄至，杀廉访副使不达失里，啖其尸。（《元史· 卷一百九十四·列传第八十一·忠义二》㉕*）
#   约1329年: （褚）不华以余兵入淮安。……城中饿者仆道上，即取啖之，一切草木、螺蛤、鱼蛙、燕乌，及靴皮、鞍韂、革箱、败弓之筋皆尽，而后父子夫妇老稚更相食，撤屋为薪，人多露处，坊陌生荆棘。力既尽，城陷。（《元史· 卷一百九十四·列传第八十一·忠义二》㉕*）
#   1328年: （天历元年十二月）陕西自泰定二年至是岁不雨，大饥，民相食。（《元史· 卷三十二·本纪第三十二·文宗一》㉕*）&lt;p&gt;天历元年八月，陕西大旱，人相食。（《元史· 卷五十·志第三上·五行一》㉕）&lt;/p&gt;
#   1329年: 天历二年，关中大旱，饥民相食。（《元史· 卷一百七十五·列传第六十二·张珪等》㉕*）&lt;p&gt;文宗天历二年三月，屯田总管兼管河渠司事郭嘉议言：“……近因奉元亢旱，五载失稔，人皆相食，流移疫死者十七八。”（《元史· 卷六十五·志第十七上·河渠二》㉕）&lt;/p&gt;&lt;p&gt;天历二年，（乃蛮台）迁陕西行省平章政事。关中大饥，……京兆民掠人而食之，则命分健卒为队，捕强食人者，其患乃已。（《元史· 卷一百三十九·列传第二十六·乃蛮台等》㉕）&lt;/p&gt;
#   1329:（天历二年夏四月）丙辰，行在所遣只儿哈郎等至京师。河南廉访司言：“河南府路以兵、旱民饥，食人肉事觉者五十一人，饿死者千九百五十人，饥者一万七千四百余人。乞弛山林川泽之禁，听民采食，行入粟补官之令，及括江淮僧道余粮以赈。”（《元史· 卷三十三·本纪第三十三·文宗二》㉕*）
#   1338年: 重改至元四年，…. 贼怒，缚景茂于树，脔其肉，使自啖。景茂益愤骂，贼遂以刀决其口，至耳傍，景茂骂不绝声而死。（《元史· 卷一百九十三·列传第八十·忠义一》㉕*）
#   1342年: 二年春正月…..，是月，大同饥，人相食，运京师粮赈之。（《元史· 卷四十·本纪第四十·顺帝三》㉕*）&lt;p&gt;至正二年，彰德、大同二郡及冀宁平晋、榆次、徐沟县，汾州孝义县，忻州皆大旱，自春至秋不雨，人有相食者。（《元史·卷五十一·志第三下·五行二》㉕）&lt;/p&gt;
#   1343年: （至正）三年，卫辉、冀宁、忻州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1344年: （至正四年）六月，河南巩县大雨，伊、洛水溢，漂民居数百家。济宁路兖州，汴梁鄢陵、通许、陈留、临颍等县大水害稼，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1344年: 八月戊午，祭社稷。丁卯，山东霖雨，民饥相食，赈之。（《元史· 卷四十一·本纪第四十一·顺帝四》㉕*）&lt;p&gt;1344年:（至正四年）八月，益都霖雨，饥民有相食者。（《元史·卷五十一·志第三下·五行二》㉕）&lt;/p&gt;
#   1345年: 五年春，东平路须城、东阿、阳谷三县及徐州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1347: 六月，……彰德路大饥，民相食。（《元史· 卷四十一·本纪第四十一·顺帝四》㉕*）
#   1348: 刘秉直，字清臣，大都武清人。至正八年，来为卫辉路总管，……岁大饥，人相食，死者过半，秉直出俸米，倡富民分粟，馁者食之，病者与药，死者与棺以葬。（《元史· 卷一百九十二·列传第七十九·良吏二》㉕*）
#   1349年: （至正）九年春，胶州大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#  [[:w:元惠宗|元惠宗]]至正年间，大饑，“淮右军”軍隊開始吃人，“天下兵甲方殷，而淮右之軍嗜食人，以小兒為上，婦女次之，男子又次之。或使坐兩缸間，外逼以火。或於鐵架上生炙。或縛其手足，先用沸湯澆潑，卻以竹帚刷去苦皮。或盛夾袋中，入巨鍋活煮。或卦作事件而淹之。或男子則止斷其雙腿，婦女則特剜其雙乳。酷毒萬狀，不可具言。總名曰「想肉」，以為食之而使人想之也。”&lt;ref&gt;{{Cite web|title=南村輟耕錄 (四部叢刊本)/卷之九 - 維基文庫，自由的圖書館|url=https://zh.wikisource.org/zh-hant/%E5%8D%97%E6%9D%91%E8%BC%9F%E8%80%95%E9%8C%84_(%E5%9B%9B%E9%83%A8%E5%8F%A2%E5%88%8A%E6%9C%AC)/%E5%8D%B7%E4%B9%8B%E4%B9%9D|website=zh.wikisource.org|access-date=2024-05-28|language=zh-Hant}}&lt;/ref&gt;
#   1352年: （至正）十二年，蕲州、黄州大旱，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1353年: 至正十二年，……明年，春夏大饥，人相食，（余阙）乃捐俸为粥以食之，得活者甚众。（《元史· 卷一百四十三·列传第三十·马祖常等》㉕*）
#   1354年: （至正）十四年，怀庆河内县、孟州，汴梁祥符县，福建泉州，湖南永州、宝庆，广西梧州皆大旱。祥符旱魃再见，泉州种不入土，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1354年: 十四年春，浙东台州，江东饶，闽海福州、邵武、汀州，江西龙兴、建昌、吉安、临江，广西静江等郡皆大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1355: 京师大饥，加以疫疠，民有父子相食者。（《元史· 卷四十三·本纪第四十三·顺帝六》㉕*）
#   1358年: 十八年春，莒州蒙阴县大饥，斗米金一斤。冬，京师大饥，人相食，彰德、山东亦如之。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1358年: （至正）十八年春，蓟州旱。莒州、滨州、般阳淄川县、霍州、鄜州、凤翔岐山县春夏皆大旱。莒州家人自相食，岐山人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1358年: 顺德九县民食蝗，广平人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1359年: （至正）十九年，大都霸州、通州，真定，彰德，怀庆，东昌，卫辉，河间之临邑，东平之须城、东阿、阳谷三县，山东益都、临淄二县，潍州、胶州、博兴州，大同、冀宁二郡，文水、榆次、寿阳、徐沟四县，沂、汾二州，及孝义、平遥、介休三县，晋宁潞州及壶关、潞城、襄垣三县，霍州赵城、灵石二县，隰之永和，沁之武乡，辽之榆社、奉元，及汴梁之祥符、原武、鄢陵、扶沟、杞、尉氏、洧川七县，郑之荥阳、汜水，许之长葛、郾城、襄城、临颍，钧之新郑、密县，皆蝗，食禾稼草木俱尽，所至蔽日，碍人马不能行，填坑堑皆盈。饥民捕蝗以为食，或曝干而积之。又罄，则人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   十九年正月至五月，京师大饥，银一锭得米仅八斗，死者无算。通州民刘五杀其子而食之。保定路莩死盈道，军士掠孱弱以为食。济南及益都之高苑，莒之蒙阴，河南之孟津、新安、黾池等县皆大饥，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   1359年: 十八年二月，江西陈友谅遣贼党王奉国等，号二十万，寇信州。明年正月，伯颜不花的斤自衢引兵援焉。……时军民唯食草苗茶纸，既尽，括靴底煮食之，又尽，掘鼠罗雀，及杀老弱以食。五月，大破贼兵。（《元史· 卷一百九十五·列传第八十二·忠义三》㉕*）
#   1360: 至正二十年，（丁好礼）遂拜中书参知政事。时京师大饥，天寿节，庙堂欲用故事大宴会，好礼言：“今民父子有相食者，君臣当修省，以弭大患，燕会宜减常度。”不听，乞谢事，乃以集贤大学士致仕，给全俸家居。（《元史· 卷一百九十六·列传第八十三·忠义四》㉕*）
#   1360年: 李仲义妻刘氏，名翠哥，房山人。至正二十年，县大饥，平章刘哈剌不花兵乏食，执仲义欲烹之。仲义弟马儿走报刘氏，刘氏遽往救之，涕泣伏地，告于兵曰：“所执者是吾夫也，乞矜怜之，贷其生，吾家有酱一瓮、米一斗五升，窖于地中，可掘取之，以代吾夫。”兵不从，刘氏曰：“吾夫瘦小，不可食。吾闻妇人肥黑者味美，吾肥且黑，愿就烹以代夫死。”兵遂释其夫而烹刘氏。闻者莫不哀之。（《元史· 卷二百一·列传第八十八·列女二》㉕*）
#   1362年:（至正）二十二年，河南洛阳、孟津、偃师三县大旱，人相食。（《元史·卷五十一·志第三下·五行二》㉕*）
#   萧道寿，京兆兴平人。……母尝有疾，医累岁不能疗，道寿刲股肉啖之而愈。至元八年，赐羊酒，表其门。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   宁猪狗，山丹州人。母年七十余，患风疾，药饵不效，猪狗割股肉进啖，遂愈。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   潭州万户移剌琼子李家奴，九岁，母病，医言不可治，李家奴割股肉，煮糜以进，病乃痊。抚州路总管管如林、浑州民朱天祥，并以母疾刲割股，旌其家。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   孔全，亳州鹿邑人。父成病，刲股肉啖之，愈。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   赵荣，扶风人。母强氏有疾，荣割股肉啖之者三。（《元史·卷一百九十七·列传第八十四·孝友一》㉕*）
#   胡伴侣，钧州密县人。其父实尝患心疾数月，几死，更数医俱莫能疗。伴侣乃斋沐焚香，泣告于天，以所佩小刀于右胁傍刲其皮肤，割脂一片，煎药以进，父疾遂瘳，其伤亦旋愈。朝廷旌表其门。（《元史· 卷一百九十八·列传第八十五·孝友二》㉕*）
#   郎氏，湖州安吉人，宋进士朱甲妻也。……家居，养姑甚谨。姑尝病，郎祷天，刲股肉进啖而愈。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   许氏女，安丰人。父疾，割股啖之乃痊。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   秦氏二女，河南宜阳人，逸其名。父尝有危疾，医云不可攻。姊闭户默祷，凿己脑和药进饮，遂愈。父后复病欲绝，妹刲股肉置粥中，父小啜即苏。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   张义妇，济南邹平人，年十八归里人李伍。……张独家居，养舅姑甚至。父母舅姑病，凡四刲股肉救不懈。（《元史·卷二百·列传第八十七·列女一》㉕*）
#   武用妻苏氏，真定人，徙家京师。用疾，苏氏刲股为粥以进，疾即愈。（《元史· 卷二百一·列传第八十八·列女二》㉕*）

==明==
[[:w:李時珍|李時珍]]完成《本草綱目》，他蒐集藥名是為了「凡經人用者，皆不可遺」，「人部」舉凡毛髮、指甲、牙齒、屎尿、唾液、乳汁、眼淚、汗水、人骨、胞衣（[[:w:紫河車|紫河車]]）、體垢、月水、人勢（陰莖）、人膽、結石……皆可入藥。頭髮可治傷寒、肚疼，男性陰毛治蛇咬，人魄（人吊死級的魂魄）可以安神定魄。

明朝没有像元朝一样法律禁止割肉疗亲，但朱元璋和其礼部尚书公开表示不赞同，但此后仍然多次出现，而且得到政府表彰，还有王族如此做，可见此风难止。

* 至（洪武）二十七年九月，山东守臣言：“日照民江伯儿，母疾，割肋肉以疗，不愈。祷岱岳神，母疾瘳，愿杀子以祀。已果瘳，竟杀其三岁儿。”帝大怒曰：“父子天伦至重。《礼》父服长子三年。今小民无知，灭伦害理，亟宜治罪。”遂逮伯儿，仗之百，遣戍海南。因命议旌表例。礼臣（任亨泰）议曰：“人子事亲，居则致其敬，养则致其乐，有疾则医药吁祷，迫切之情，人子所得为也。至卧冰割股，上古未闻。倘父母止有一子，或割肝而丧生，或卧冰而致死，使父母无依，宗祀永绝，反为不孝之大。皆由愚昧之徒，尚诡异，骇愚俗，希旌表，规避里徭。割股不已，至于割肝，割肝不已，至于杀子。违道伤生，莫此为甚。自今父母有疾，疗治罔功，不得已而卧冰割股，亦听其所为，不在旌表例。”制曰：“可。”（《明史·卷一百三十七·列传第二十五·刘三吾等》）

食人事件的记载：

#  [[:w:韩观|韩观]]杀人甚多，御史欲弹劾他。一日，观召御史饮，以人皮为坐褥，耳目口鼻显然，发散垂褥，首披椅后。肴上，设一人首，观以箸取二目食之，曰：“他禽兽目皆不可食，惟人目甚美。”观前席坐，每拿人至，命斩之，不回首视，已而血流满庭。观曰： “此辈与禽兽不异，斩之如杀虎豹耳。”御史战栗失措曰：“公，神人也。”竟不能劾。&lt;ref&gt;《[[s:湧幢小品/09#韓都督應變|湧幢小品 韓都督應變]]》朱国桢&lt;/ref&gt;
#  1385年，洪武十八年：（韩）林儿本起盗贼，无大志，又听命福通，徒拥虚名。诸将在外者率不遵约束，所过焚劫，至啖老弱为粮，且皆福通故等夷，福通亦不能制。（《明史·卷一百二十二·列传第十·郭子兴 韩林儿》㉕*）
#  约1426年，宣德年间：得（朱）有熺掠食生人肝脑诸不法事，于是并免为庶人。（《明史·卷一百十六·列传第四·诸王》㉕*）
#  1454年，景泰五年：景泰五年，广西古丁等洞贼首蓝伽、韦万山等，纠合蛮类，劫掠南宁、上林、武缘诸处。……贼首韦朝威据古田，县官窜会城，遣典史入县抚谕，烹食之。（《明史·卷三百十七·列传第二百五·广西土司》㉕*）
#  1457年，天顺元年：北畿、山東並飢，發塋墓，斫道樹殆盡。父子或相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  约1465年，成化初：成化初，（彭伦）从赵辅，平大藤峡贼。……（彭）伦大会所部目、把缚俘囚，置高竿，集健卒乱射杀之，复割裂肢体，烹啖诸壮士。（《明史·卷一百六十六·列传第五十四·韩观等》㉕*）
#  1484年，成化二十年：是秋，陝西、山西大旱饑，人相食。停歲辦物料，免稅糧，發帑轉粟，開納米事例振之。（《明史·卷十四·本纪第十四·宪宗二》㉕*）&lt;p&gt;又有虎臣者，麟游人。成化中贡入太学。……省亲归，会陕西大饥，……上言：“臣乡比岁灾伤，人相食，由长吏贪残，赋役失均。请敕有司审民户，编三等以定科徭。”从之。（《明史·卷一百六十四·列传第五十二·邹缉等》㉕）&lt;/p&gt;&lt;p&gt;十六年（何乔新）擢右副都御史，巡抚山西。……进左副都御史。……召拜刑部右侍郎。山西大饥，人相食。命往振，活三十余万人，还流冗十四万户。（《明史·卷一百八十三·列传第七十一·何乔新等》㉕）&lt;/p&gt;&lt;p&gt;汪奎，字文灿，婺源人。……（成化）二十一年，星变，偕同官疏陈十事，言：“……山、陕、河、洛饥民多流郧、襄，至骨肉相啖。请大发帑庾振济，消弭他变。”（《明史·卷一百八十·列传第六十八·张宁等》㉕）&lt;/p&gt;
#  1504年，弘治十七年：十七年，淮、扬、庐、凤洊饥，人相食，且发瘗胔（坟墓尸体）以继之。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1518年，正德十三年：佛郎机，近满剌加。正德中，据满剌加地，逐其王。十三年遣使臣加必丹末等贡方物，请封，始知其名。诏给方物之直，遣还。其人久留不去，剽劫行旅，至掠小儿为食。（《明史·卷三百二十五·列传第二百十三·外国六》㉕*）
#  正德五年（1510年）八月，[[:w:刘瑾|刘瑾]]被磔死，凌迟三日，共剐3300余刀。行刑之日，北京鼎沸，百姓相爭以一钱买刘瑾一塊肉，生吞以泄恨。{{Citation needed}}
#  1519年，正德十四年：是岁，淮、扬饥，人相食。（《明史·卷十六·本纪第十六·武宗》㉕*）&lt;p&gt;十四年三月，有诏南巡，（黄）巩上疏曰：……今江、淮大饥，父子兄弟相食。（《明史·卷一百八十九·列传第七十七·李文祥等》㉕）&lt;/p&gt;&lt;p&gt;（吴）一鹏极陈四方灾异，言：“自去年六月迄今二月，其间天鸣者三，地震者三十八，秋冬雷电雨雹十八，暴风、白气、地裂、山崩、产妖各一，民饥相食二。非常之变，倍于往时。愿陛下率先群工，救疾苦，罢营缮，信大臣，纳忠谏，用回天意。”（《明史·卷一百九十一·列传第七十九·毛澄等》㉕）&lt;/p&gt;
#  1524年，嘉靖三年：三年，湖广、河南、大名、临清饥。南畿诸郡大饥，父子相食，道殣相望，臭弥千里。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;（张）汉卿言：“……今东南洊饥，民至骨肉相食，而搜括之令频行，臣等窃以为不可。”（《明史·卷一百九十二·列传第八十·杨慎》㉕）&lt;/p&gt;&lt;p&gt;世宗即位，（韩邦靖）起山西左参议，分守大同。岁饥，人相食，奏请发帑，不许。（《明史·卷二百一·列传第八十九·陶琰等》㉕）&lt;/p&gt;&lt;p&gt;嘉靖四年二月（余珊）应诏陈十渐，其略曰：……近年以来，黄纸蠲放，白纸催征；额外之敛，下及鸡豚；织造之需，自为商贾。江、淮母子相食，兖、豫盗贼横行，川、陕、湖、贵疲于供饷。（《明史·卷二百八·列传第九十六·张芹等》㉕）&lt;/p&gt;&lt;p&gt;嘉靖初，（湛若水）入朝，……明年进侍读，复疏言：“一二年间，天变地震，山崩川涌，人饥相食，殆无虚月。”（《明史·卷二百八十三·列传第一百七十一·儒林二》㉕）&lt;/p&gt;
#  1529年，嘉靖八年：（杨爵）登嘉靖八年进士，授行人。帝方崇饰礼文，（杨）爵因使王府还，上言：“臣奉使湖广，睹民多菜色，挈筐操刃，割道殍食之。（《明史·卷二百九·列传第九十七·杨最等》㉕*）
#  1549年，嘉靖二十八年：有吴国佐者，洪州司特峒寨苗也，….. 其党石纂太称“太保”，合攻上黄堡，诱败参将黄冲霄，追至永从县，杀守备张世忠，炙而啖之。（《明史·卷二百四十七·列传第一百三十五·刘綎等》㉕*）
#  1552年，嘉靖三十一年：宣、大二镇大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1553年，嘉靖三十二年：京师大饥，人相食，米石二两二钱。（《历代社会风俗事物考》引《金垒子》）
#   1557年，嘉靖三十六年：三十六年，辽东大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1559年，嘉靖三十八年八月：以辽东连年饥馑，至有父食死子者，发银糴粟赈之。（《中外历史年表》）
#   1588，万历十六年：十六年，河南饥，民相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#   1591年，万历十九年：（万历）十九年，（子俊民）还理部事。河南大饥，人相食，请发银米各数十万。（《明史·卷二百十四·列传第一百二·杨博等》㉕*）
#   1593年，万历二十二年：二十二年，河南大饥，人相食，命（钟）化民兼河南道御史往振。荒政具举，民大悦。（《明史·卷二百二十七·列传第一百十五·庞尚鹏等》㉕*）&lt;/p&gt;&lt;p&gt;（陈登云）出按河南。岁大饥，人相食。（《明史·卷二百三十三·列传第一百二十一·姜应麟等》㉕）&lt;/p&gt;
#   1601年，万历二十九年：二十九年，两畿饥。阜平县饥，有食其稚子者。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1611年，万历三十九年：马孟祯，字泰符，桐城人。万历二十六年进士。……三十九年夏，怡神殿灾。孟祯言：“二十年来，郊庙、朝讲、召对、面议俱废，通下情者惟章奏。……畿辅、山东、山西、河南，比岁旱饥。民间卖女鬻儿，食妻啖子，铤而走险，急何能择。”（《明史·卷二百三十·列传第一百十八·蔡时鼎等》㉕*）
#  康熙十二年修《青州府志》第20卷载，万历四十三年（1615年），山东青州府推官[[:w:黄槐开|黄槐开]]在一件申文中说：“自古饥年，止闻道殣相望与易子而食、析骸而爨耳。今屠割活人以供朝夕，父子不问矣，夫妇不问矣，兄弟不问矣。剖腹剜心，支解作脍，且以人心味为美，小儿味尤为美。甚有鬻人肉于市，每斤价钱六文者；有腌人肉于家，以备不时之需者；有割人头用火烧熟而吮其脑者；有饿方倒而众刀攒割立尽者；亦有割肉将尽而眼瞪瞪视人者。间有为人所诃禁，辄应曰：&quot;我不食人，人将食我。&quot;愚民恬不为怪，有司法无所施。枭獍在途，天地昼晦。”
#  1616年，万历四十四年：四十四年，山东饥甚，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;今春以来，天鼓两震于晋地，流星昼陨于清丰，地震二十八，天火九，石首雨菽，河内女妖，辽东兵端吐火，即春秋二百四十年间，未有稠于今日者。且山东大昆，人相食，黄河水稽天。（《明史·卷二百五十七·列传第一百四十五·张鹤鸣等》㉕）&lt;/p&gt;&lt;p&gt;“以山东大饥，致母食死儿，夫食死妻，再振之。”（《中外历史年表》）&lt;/p&gt;
#  萬曆四十五年（1617年）連兩年山東大饑，蔡州有人肉市。“中州兄弟两无子，去山东买妾，遇二女，自称姑嫂，骗兄弟往。兄得小姑。小姑私语之曰：汝弟已为我嫂制成肉羹矣。兄急往视，弟头尚扔炕下。兄急诉之县，抵嫂于罪，兄带小姑去。”（《[[:w:棗林杂俎|棗林杂俎]]》）
#  近日福建抽稅太監高采謬聽方士言：食小兒腦千餘，其陽道可復生如故。乃遍買童稚潛殺之。久而事彰聞，民間無肯鬻者，則令人遍往他所盜至送入，四方失兒者無算，遂至激變掣回。此等俱飛天夜叉化身也。&lt;ref&gt;[[s:萬曆野獲編/卷28#食人]]&lt;/ref&gt;
#  约1621年，天启初：天启初，奢崇明反，率众薄城。（董）尽伦偕知州翁登彦固守。贼遣使说降，尽伦大怒，手刃贼使，抉其睛啖之，屡挫贼锋，城获全。（《明史·卷二百九十·列传第一百七十八·忠义二》㉕*）
#  1622年，天启二年：万化亦率苗仲九股陷龙里，遂围贵阳，自称罗甸王，时天启二年二月也。……外援既绝，攻益急，城中粮尽，人相食，而拒守不遗余力。（《明史·卷三百十六·列传第二百四·贵州土司》㉕*）&lt;p&gt;方官廪之告竭也，米升直二十金。食糠核草木败革皆尽，食死人肉，后乃生食人，至亲属相啖。彦方、运清部卒公屠人市肆，斤易银一两。枟尽焚书籍冠服，预戒家人，急则自尽，皆授以刀缳。城中户十万，围困三百日，仅存者千余人。（《明史·卷二百四十九·列传第一百三十七·朱燮元等》㉕）&lt;/p&gt;
#  1627年，清皇太极之天聪元年，天启七年：（清）国中大饥，斗米价银八两（天启时金一两合銀十两），人有相食者。国中银两虽多，无外贸易，是以银贱而诸物腾贵。良马银300两，牛一银百两，蟒缎一，银百五十两，布一匹，银九两。（《清太宗实录卷三》）
#  “天启辛酉，延安、庆阳、平凉旱，岁大饥。东事孔棘，有司惟顾军兴，征督如故，民不能供，道殣相望。或群职富者粟，惧捕诛，始聚为盗。盗起，饥益甚，连年赤地，斗米千钱不能得，人相食，从乱如归。饥民为贼由此而始。”&lt;ref&gt;《怀陵流寇始终录》，卷1，1页。&lt;/ref&gt;
#  1629年，崇禎二年，殺[[:w:袁崇煥|袁崇煥]]。[[:w:張岱|張岱]]《石匱書後集》：“（袁崇煥）遂於鎮撫司綁發西市，寸寸臠割之。割肉一塊，京師百姓從劊子手爭取生啖之。劊子亂撲，百姓以錢爭買其肉，頃刻立荊開腔出其腸胃，百姓群起搶之，得其一節者，和燒酒生嚙，血流齒頰間，猶唾地罵不已。拾得其骨者，以刀斧碎磔之，骨肉俱盡，止剩一首，傳視九邊。”，“时百姓怨恨，争啖其肉，皮骨已尽，心肺之间犹叫声不绝，半日而止，所谓活剐者也……百姓将银一钱，买肉一块，如手指大，噉之。食时必骂一声，须臾崇焕肉悉卖尽。”（[[:w:计六奇|计六奇]]：《[[:w:明季北略|明季北略]]》卷五）
#  1633年，崇祯六年：（陈）三接，文水人。举崇祯六年乡试，知河间县。岁旱饥，人相食。（《明史·卷二百九十一·列传第一百七十九·忠义三》㉕*）
#  1634年，崇祯七年：七年，京师饥，御史龚廷献绘《饥民图》以进。太原大饥，人相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;p&gt;七年，西北大旱，秦、晋人相食，（吴甘来）疏请发粟以振。（《明史·卷二百六十六·列传第一百五十四·马世奇等》㉕）&lt;/p&gt;
#  1636年，崇祯九年：山西大饥，人相食。（《明史·卷二十三·本纪第二十三·庄烈帝一》㉕*）
#   1637年，崇祯十年：十年浙江大饥，父子、兄弟、夫妻相食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  崇禎十二年（1639年）[[:w:鄭鄤|鄭鄤]]以「杖母、姦妹」罪被磔死。《[[:w:明季北略|明季北略]]》记载鄭鄤被凌迟三千六百刀後，为“都人士”药用：“炮声响后，人皆跻足引领，顿高尺许，拥挤之极……归途所见，买生肉为疮疥药料者，遍长安市。二十年前之文章气节，功名显宦，竟与参术甘皮同奏肤功，亦大奇也。”
#  1639年，崇祯十二年：十二年，两畿、山东、山西、陕西、江西饥。河南大饥，人相食，卢氏、嵩、伊阳三县尤甚。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）
#  1640年，崇禎十三年，全國有123州縣發生“人相食”，98州縣蝗災。{{Citation needed}}&lt;p&gt;是年，两畿、山东、河南、山、陕旱蝗，人相食。（《明史·卷二十四·本纪第二十四·庄烈帝二》㉕*）&lt;/p&gt;&lt;p&gt;关河大旱，人相食，土寇蜂起，陕西窦开远、河南李际遇为之魁，饥民从之，所在告警。（《明史·卷二百五十二·列传第一百四十·杨嗣昌等》㉕）&lt;/p&gt;&lt;p&gt;十三年，北畿、山东、河南、陕西、山西、浙江、三吴皆饥。自淮而北至畿南，树皮食尽，发瘗胔（坟墓里的尸体）以食。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕）&lt;/p&gt;
#  1641年，崇祯十四年：德州斗米千钱，父子相食，行人断绝。大盗滋矣。（《明史·卷三十·志第六·五行三（金 土）·年饥》㉕*）&lt;/p&gt;&lt;p&gt;及崇祯时，常洵地近属尊，朝廷尊礼之。常洵日闭阁饮醇酒，所好惟妇女倡乐。秦中流贼起，河南大旱蝗，人相食，民间藉藉，谓先帝耗天下以肥王，洛阳富于大内。（《明史·卷一百二十·列传第八·诸王五》㉕）&lt;/p&gt;&lt;p&gt;芳奕，慷慨负智略，与秉衡同举于乡，为昌乐知县。解官归，岁大歉，人相食，倾橐济之。（《明史·卷二百九十三·列传第一百八十一·忠义五》㉕）&lt;/p&gt;&lt;p&gt;十四年（左懋第）督催漕运，道中驰疏言：“臣自静海抵临清，见人民饥死者三，疫死者三，为盗者四。米石银二十四两，人死取以食，惟圣明垂念。”（《明史·卷二百七十五·列传第一百六十三·张慎言等》㉕）&lt;/p&gt; 崇禎十四年（1641年），「浙江大旱，飛蝗蔽天，食草根幾盡，人饑且疫」。崇祯十四年二月，时山东荒旱，寇盗益炽，徐德（南端到北端）数千里-{}-白骨纵横，父子相食，人迹断绝。（彭贻孙《平寇志》）
#  1641年，崇祯十四年：（九月）十一日，秦师食尽，宗龙杀马骡以享军。明日，营中马骡尽，杀贼取其尸分啖之。（《明史·卷二百六十二·列传第一百五十·傅宗龙等》㉕*）
#  明朝末年，四川大饑，“蜀大飢，人相食。先是丙戌、丁亥，連歲干涸，至是彌甚。赤地千里，糲米一斗價二十金，養麥一斗價七八金，久之亦無賣者篙芹木葉，取食殆盡。時有裹珍珠二昇，易一面不得而殆：有持數百金，買一飽不得而死。於是人皆相食，道路飢殍，剝取殆盡。無所得，父子、兄弟、夫妻，轉相賊殺。”（清·彭遵泅《蜀碧》卷四）
#  「庚辰山西大饑，人相食，剖心，其竅多寡不等。或無竅，或五六，其二、三竅為多，心大小各異。」（《[[:w:棗林雜俎|棗林雜俎]]·和集》）
#  明朝崇禎末年，河南和山東發生饑荒和蝗災，可以吃的東西都已經吃完，唯一剩下的可以吃的就只有人，於是便有了公開的人肉市場，其販賣的乃是活生生的人，稱之曰“[[:w:菜人|菜人]]”。[[:w:紀昀|紀昀]]《[[:w:閱微草堂筆記|閱微草堂筆記]]》有這樣的記載：“婦女幼孩，反接鬻於市，謂之菜人”。&lt;ref&gt;{{cite wikisource |title=《閱微草堂筆記》 |wslink=閱微草堂筆記 |chapter=卷2 |author=紀昀 | authorlink=紀昀}}&lt;/ref&gt;而在[[:w:屈大均|屈大均]]創作的一首七言古詩《[[s:菜人哀|菜人哀]]》，內容即以第一視角描述一對夫妻在崇禎末年，一位丈夫因過於飢餓，將妻子賣予一家屠戶成為“菜人”。
#  《陕西通志》第86卷载有明朝崇祯年间[[:w:马懋才|马懋才]]的《备陈灾变疏》，疏中写道：“臣乡延安府，自去岁一年无雨，草木枯焦。八、九月间，民争采山间蓬草而食，其粒类糠皮，其味苦而涩，食之仅可延以不死。至十月以后而蓬尽矣；则剥树皮而食。诸树惟榆树差善，杂他树皮以为食，亦可稍缓其死。殆年终而树皮又尽矣，则又掘山中石块而食。甘石名青叶，味腥而腻，少食辄饱，不数日则腹胀下坠而死。民有不甘于食石以死者始相聚为盗，而一、二稍有积贮之民遂为所劫，而抢掠无遗矣。有司亦不能禁治。间有获者亦恬不知畏；且曰：“死于饥与死于盗等耳，与其坐而饥死，何若为盗而死，犹得为饱鬼也。”
#  [[:w:計六奇|計六奇]]說：“天降奇荒，所以资自成也！”&lt;ref&gt;{{cite wikisource |title=《明季北略》 |wslink=明季北略 |chapter=卷05 |author=計六奇|authorlink=計六奇}}&lt;/ref&gt;。 
#  崇禎十四年（1641年）二月，[[:w:李自成|李自成]]攻陷洛陽，殺重達三百六十多斤的福王[[:w:朱常洵|朱常洵]]，用他的肉和皇家園林裡的[[:w:梅花鹿|梅花鹿]]一同烹煮，在洛陽西關周公廟舉行宴會，賜給部下食用，名曰“福祿宴”。&lt;ref&gt;《明季北略·卷十七》：王体肥，重三百余筋，贼置酒大会，以王为菹，杂鹿肉食之，号福禄酒。&lt;/ref&gt;
#  约1644年，顺治二年：（刘）泽清颇涉文艺，好吟咏。尝召客饮酒唱和。幕中蓄两猿，以名呼之即至。一日，宴其故人子，酌酒金瓯中，瓯可容三升许，呼猿捧酒跪送客。猿狰狞甚，客战掉，逡巡不敢取。泽清笑曰：“君怖耶？”命取囚扑死阶下，剜其脑及心肝，置瓯中，和酒，付猿捧之前。饮酹，颜色自若。其凶忍多此类。（《明史·卷二百七十三·列传第一百六十一·左良玉等》㉕*）
#   明末：中原盗起十余年，所在荼毒，督抚莫能办，率倡抚议，苟且幸无事，盗且服且叛。而河南比年大旱蝗，人相食，民益蜂起为盗。（《清史稿·卷五百·列传二百八十七·遗逸一》㉕*）
#  时有将军安氵侃者，一岁丧母，事其父以孝闻。父病革，刲臂为汤饮父，父良已。（《明史·卷一百十六·列传第四·诸王》㉕*）
#  襄陵王冲秌，宪王第二子，有至性。母病，刲股和药，病良已。（《明史·卷一百十八·列传第六·诸王三》㉕*）
# （襄陵王冲秌之）子范址服其教，母荆罹危疾，亦刲股进之，愈。（《明史·卷一百十八·列传第六·诸王三》㉕*）
#  刘铉，字宗器，长洲人。生弥月而孤。及长，刲股疗母疾。母卒，哀毁，以孝闻。（《明史·卷一百六十三·列传第五十一·李时勉等》㉕*）
# （孙）祖寿初守固关，遘危疾，妻张氏割臂以疗，绝饮食者七日。祖寿生，而张氏旋死，遂终身不近妇人。（《明史·卷二百七十一·列传第一百五十九·贺世贤》㉕*）
#  朱鉴，字用明，晋江人。童时刲股疗父疾。举乡试，授蒲圻教谕。（《明史·卷一百七十二·列传第六十·罗亨信等》㉕*）
#  储巏，字静夫，泰州人。九岁能属文。母疾，刲股疗之，卒不起。（《明史·卷二百八十六·列传第一百七十四·文苑二》㉕*）
#  许琰，字玉仲，吴县人。幼有至性，尝刲臂疗父疾。（《明史·卷二百九十五·列传第一百八十三·忠义七》㉕*）
#  沈德四，直隶华亭人。祖母疾，刲股疗之愈。己而祖父疾，又刲肝作汤进之，亦愈。洪武二十六年被旌。寻授太常赞礼郎。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  上元姚金玉、昌平王德儿亦以刲肝愈母疾，与德四同旌。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  至二十七年九月，山东守臣言：“日照民江伯儿，母疾，割肋肉以疗，不愈。祷岱岳神，母疾瘳，愿杀子以祀。已果瘳，竟杀其三岁儿。”帝大怒曰：“父子天伦至重。《礼》父服长子三年。今小民无知，灭伦害理，亟宜治罪。”遂逮伯儿，仗之百，遣戍海南。因命议旌表例。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  永乐间，江阴卫卒徐佛保等复以割股被旌。（《明史·卷二百九十六·列传第一百八十四·孝义》㉕*）
#  夏子孝，字以忠，桐城人。六岁失母，哀哭如成人。九岁父得危疾，祷天地，刲股六寸许，调羹以进，父食之顿愈。翌日，子孝痛创，父诘其故，始知之。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  金子良亦有孝行，父病，刲股为羹以进，旋愈。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  唐俨，全州诸生也。父荫，郴州知州，归老得危疾。俨年十二，潜割臂肉进之，疾良已。及父殁，哀毁如成人。其后游学于外，嫡母寝疾。俨妻邓氏年十八，奋曰：“吾妇人，安知汤药。昔夫子以臂肉疗吾舅，吾独不能疗吾姑哉？”于是割胁肉以进，姑疾亦愈。（《明史·卷二百九十七·列传第一百八十五·孝义二》㉕*）
#  刘孝妇，新乐韩太初妻。……刘事姑谨，姑道病，刺血和药以进。……及姑疾笃，刲肉食之，少苏，逾月而卒，殡之舍侧。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  程氏，扬州胡尚絅妻。尚絅婴危疾，妇刲腕肉啖之，不能咽而卒。妇号恸不食二日。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  杨泰奴，仁和杨得安女。许嫁未行。天顺四年，母疫病不愈。泰奴三割胸肉食母，不效。一日薄幕，剖胸取肝一片，昏仆良久。及苏，以衣裹创，手和粥以进，母遂愈。母宿有膝挛疾，亦愈。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  后有张氏，仪真周祥妻。姑病，医百方不效。一方士至其门曰：“人肝可疗。”张割左胁下，得膜如絮，以手探之没腕，取肝二寸许，无少痛，作羹以进姑，病遂瘳。（《明史·卷三百一·列传第一百八十九·列女一》㉕*）
#  李孝妇，临武人，名中姑，适江西桂廷凤。姑邓患痰疾，将不起，妇涕泣忧悼。闻有言乳肉可疗者，心识之。一日，煮药，巘香祷灶神，自割一乳，昏仆于地，气已绝。廷凤呼药不至，出视，见血流满地，大惊呼救，倾骇城市，邑长佐皆诣其庐，命亟治。俄有僧踵门曰：“以室中蕲艾傅之，即愈。”如其言，果苏，比求僧不复见矣。乃取乳和药奉姑，姑竟获全。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  洪氏，怀宁章崇雅妻。崇雅早卒，洪守志十年。姑许，疾不能起，洪剜乳肉为羹而饮之，获愈，余肉投池中，不令人知。数日后，群鸭自水中衔出，鸣噪回翔，小童获以告姑。姑起视之，乳血犹淋漓也。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  倪氏，兴化陆鳌妻。性纯孝，舅早世，悯姑老，朝夕侍寝处，与夫睽异者十五年。姑鼻患疽垂毙，躬为吮治，不愈，乃夜焚香告天，割左臂肉以进，姑啖之愈。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  刘氏，张能信妻，太仆卿宪宠女，工部尚书九德妇也。性至孝，姑病十年，侍汤药不离侧。及病剧，举刀刲臂，侍婢惊持之。舅闻，嘱医言病不宜近腥腻，力止之。逾日，竟刲肉煮糜以进，则乃姑已不能食，乃大悔恨曰：“医绐我，使姑未鉴我心。”复刲肉寸许，恸哭奠箦前，将阖棺，取所奠置棺中曰：“妇不获复事我姑，以此肉伴姑侧，犹身事姑也。”乡人莫不称其孝。（《明史·卷三百二·列传第一百九十·列女二》㉕*）
#  （颍）州又有台氏，诸生张云鹏妻。夫病，氏单衣蔬食，祷天愿代，割臂为糜以进。（《明史·卷三百三·列传第一百九十一·列女三》㉕*）

==清==
《清史稿》记载的割肉疗亲的事迹比二十五史以往各朝都多，但其实雍正有一段诏书不赞同割肉疗亲，朝廷的实际做法似乎是迫于民情不得已的情况下低调褒奖（“破格报可”），社会风气看来是称赞这种行为的。

* 雍正六年三月壬子，世宗谕曰：“……父母爱子，无所不至，若因己病而致其子割肝刲股以充饮馔、和汤药，纵其子无恙，父母未有不惊忧恻怛惨惕而不安者，况因此而伤生，岂父母所忍闻乎？父母有疾，固人子尽心竭力之时，傥能至诚纯孝，必且感天地、动鬼神，不必以惊世骇俗之为，著奇于日用伦常之外。……倘训谕之后，仍有不爱躯命，蹈于危亡者，朕亦不概加旌表，以成激烈轻生之习也。”（《清史稿·卷四百九十七·列传二百八十四·孝义一》）
* 清兴关外，俗纯朴，爱亲敬长，内悫而外严。既定鼎，礼教益备。定旌格，循明旧。亲存，奉侍竭其力；亲殁，善居丧，或庐于墓；亲远行，万里行求，或生还，或以丧归。友于兄弟，同居三五世以上，号义门，及诸义行，皆礼旌。亲病，刲股刳肝；亲丧，以身殉：皆以伤生有禁，有司以事闻，辄破格报可。所以教民者，若是其周其密也。国史承前例，撰次孝友传，亦颇及诸义行。（《清史稿·卷四百九十七·列传二百八十四·孝义一》）

历史记载中清朝的食人事件：

#  努尔哈赤时代：扬古利，舒穆禄氏，世居浑春。父郎柱，为库尔喀部长，率先附太祖，……扬古利手刃杀父者，割耳鼻生啖之，时年甫十四，太祖深异焉。（《清史稿·卷二百二十六·列传十三·扬古利等》㉕*）
#  清初：虞尔忘、尔雪，江南无锡人。国初江南多盗，尔忘、尔雪父罕卿董乡团，……罕卿死桥下矣。……知为盗杜息（所杀）。….. 比明，尔忘抱罕卿木主至，尔雪于其旁爇釜，尔忘取（杜）息舌，尔雪探心肝，且祭且啖，尔忘乃断（杜）息头。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  1627年，天聪元年，《太宗实录卷三》：“时国中大饥，斗米价银八两，人有相食者。国中银两虽多，无外贸易，是以银贱而诸物腾贵。良马，银三百两。牛一，银百两。蟒缎一，银百五十两。布匹一，银九两。盗贼繁兴，偷窃牛马，或行劫杀。于是诸臣入奏曰：盗贼若不按律严惩，恐不能止息。上恻然，谕曰：今岁国中因年饥乏食，致民不得已而为盗耳。缉获者，鞭而释之可也。遂下令，是岁谳狱，姑从宽典。仍大发帑金，散赈饥民。”
#  1631年，皇太极天聪四年：顷大凌河之役，城中人相食，明人犹死守，及援尽城降，而锦州、松、杏犹不下。（《清史稿·卷二·本纪二·太宗本纪一》㉕*）&lt;p&gt;旋有王世龙者，越城出降，言城中粮竭，商贾诸杂役多死，存者人相食，马毙殆尽。（《清史稿·卷二百三十四·列传二十一·孔有德等》㉕）&lt;/p&gt;&lt;p&gt;祖大壽疏奏：“被圍將及三月，城中食盡，殺人相食。”（《崇禎長編》卷五二）。&lt;/p&gt;&lt;p&gt;明大凌河城內，糧絕薪盡。軍士飢甚，殺其修城夫役及商賈平民為食，析骸而炊。又執軍士之羸弱者，殺而食之。（《清太宗實錄·卷十》）&lt;/p&gt;
#  1635年，皇太极天聪八年：先是，察哈尔林丹西奔图白特，其部众苦林丹暴虐，逗遛者什七八，食尽，杀人相食，屠劫不已，溃散四出。（《清史稿·卷二·本纪二·太宗本纪一》㉕*）
#  1645年，顺治二年：二年，耒（枣？）阳、襄阳、光化、宜城大饥，人相食。”（{{cite wikisource |title=《清史稿·卷44·志十九·災異五》 |wslink=清史稿/卷44 |author=趙爾巽|authorlink=趙爾巽}}㉕*）
#  1648年，顺治五年：五年春，广州、鹤庆（大理，洱海之北）嵩明（昆明市东北）大饥，人相食。”（{{cite wikisource |title=《清史稿·卷42·志十七·災異三》 |wslink=清史稿/卷42 |author=趙爾巽|authorlink=趙爾巽}}㉕*）
#  順治九年八月，漳州被圍半年，城中缺糧，一碗稀粥索價白銀四兩。居民以老鼠、麻雀、樹根、樹葉、水萍、紙張和皮革等物為食，餓死者不計其數，“城中人自相食，百姓十死其八，兵馬盡皆枵腹”&lt;ref&gt;《明清史料》丁編，第一本，第七十五頁《查明漳州解圍功次殘件》。&lt;/ref&gt;。
#  1654年，顺治十一年：顺治十一年，明将李定国攻新会，城守阅八月，食尽，杀人马为食。（《清史稿·卷五百十·列传二百九十七·列女三》㉕*）
#  顺治年間，“安邑知县鹿尽心者，得痿痺疾。有方士挟乩术，自称刘海蟾，教以食小儿脑即愈。鹿信之，辄以重价购小儿击杀食之，所杀伤甚众，而病不减。因复请于乩仙，复教以生食乃可愈。因更生凿小儿脑吸之。致死者不一，病竟不愈而死。事随彰闻，被害之家，共置方士于法。”&lt;ref&gt;[[:w:王士祯|王士祯]]：《池北偶谈·鹿尽心》&lt;/ref&gt;
#  康熙十八年（1679年），山东“终年不雨，大饥，人相食。”（乾隆《青城（即今高青）县-{}-志》卷10）
#  1681年，康熙二十年：诇知粮将罄，人相食，与诸将环而攻之。（吴）世璠众内乱，欲擒世璠以降，世璠自杀。（《清史稿·卷二百五十四·列传四十一·赉塔等》㉕*）
#  1698年，康熙三十七年春：三十七年春，平定、乐平大饥，人相食。”（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1703年，康熙四十二年：永年（邯郸东北）、东明（大名府之南部，山东曹州西）饥。秋：沛县、亳州、东阿、曲阜、蒲县（属隰州，非蒲城县）、滕县大饥。冬，汶上、沂州、莒州、兖州、东昌、郓城大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1704年，康熙四十三年：四十三年春，泰安大饥，人相食，死者枕藉。肥城，东平大饥，人相食。武定（惠民）、滨州（武定东）、商河（武定西南）、阳信（武定北）、利津、沾化饥；兖州、登州大饥，民死大半，至食屋草；昌邑、即墨、掖县、高密、膠州大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#   1785年，乾隆五十年：秋，寿光、昌乐、安丘、诸城大饥，父子相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1786年，乾隆五十一年：五十一年春，山东各府、州、县大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）&lt;p&gt;《病榻梦痕录》卷上乾隆五十一年（１７８６）条记载了苏皖鲁等地的灾情，时灾民卖妻鬻子，“流丐载道”，“尸横道路”，尸体“埋于土，辄被人刨发，刮肉而啖”。&lt;/p&gt;
#  1801，嘉庆六年: 罗思举，字天鹏，四川东乡人。……（嘉庆）六年，歼张世龙于铁溪河，……自是转战老林，饷不时至，煮马鞯，啗贼肉以追贼。……尝酒酣袒身示人，战创斑斑，为父母刲股痕凡七，其忠孝盖出天性云。（《清史稿·卷三百四十七·列传一百三十四·杨遇春等》㉕*）
#  1832年，道光十二年：夏，紫阳大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1833年，道光十三年：夏，保康、郧县、房县饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1834年，道光十四年：十四年春，归州、兴山大饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1847年，道光二十七年：二十七年，南乐饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1857年，咸丰七年：七年春，肥城、东平大饥，死者枕藉；鱼台、日照、临朐亦饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1863年，同治二年，[[:w:石達開|石達開]]的軍隊為[[:w:大渡河|大渡河]]的涨水所阻，當時石部全軍已是“覓食無所得，有相殺噬人肉者”。（许亮儒遗著《擒石野史》）
#  [[:w:陈康祺|陈康祺]]《郎潜纪闻二笔》记载“同治三、四年，皖南到处食人，人肉始买三十文一斤，后增至一百二十文一斤，句容、二溧，八十文一斤，惨矣。”
#  同治三年（1864年），皖南人相食，人肉價格大漲。《曾国藩日记》同治三年四月廿二日记载：“皖南到处食人，人肉始卖三十文一斤，近闻增至百二十文一斤，句容、二溧八十文一斤。”《曾國藩日記》又記載：“[[:w:太平天国|洪楊]]之亂，[[:w:江蘇|江蘇]]人肉賣九十文一斤，漲到一百三十文錢一斤。”
#  1866年，同治五年：五年，兰州饥，人相食。（《清史稿·卷四十四·志十九·灾异五》㉕*）
#  1867年，同治六年：五年，（穆图善）收灵州。……明年，署陕甘总督，值岁大饥，人相食。（《清史稿·卷四百五十四·列传二百四十一·刘锦棠》㉕*）
#  1868年，同治七年：七年春，即墨、孝义厅、蓝田、沔县饥。夏，泾州大饥，人相食。《清史稿·卷四十四·志十九·灾异五》㉕*）&lt;p&gt;时庆阳大饥，人相食。（《清史稿·卷四百五十四·列传二百四十一·刘锦棠等》㉕）&lt;/p&gt;&lt;p&gt; 同治七年（1868年），[[:w:定西|定西]]、[[:w:通渭|通渭]]大旱，時逢戰亂，瘟疫並起，人相食。{{Citation needed|Date=January 2025}}&lt;/p&gt;
#  1877年，光绪三年：是岁，山、陕大旱，人相食。（《清史稿·卷二十三·本纪二十三·德宗本纪一》㉕*）&lt;p&gt;丁戊奇荒是中国华北地区发生于清朝光绪元年（1875年）至四年（1878年）之间的一场罕见的特大旱灾饥荒。灾害波及山西、直隶、陕西、河南、山东、甘肃等好几个省份，“饿殍载途，白骨盈野”，饿死的人竟达一千万以上，逃亡两千万以上。随著灾情的发展，可食之物的罄尽，“人食人”的惨剧发生了。大旱的第三年（1877年）冬天，重灾区山西，到处都有人食人现象。吃人肉、卖人肉者，比比皆是。有活人吃死人肉的，还有将老人或孩子活杀吃的……无情旱魔，把灾区变成了人间地狱! 在河南，侥幸活下来的饥民大多奄奄一息，“既无可食之肉，又无割人之力”，一些气息犹存的灾民，倒地之后即为饿犬残食。{{Citation needed|Date=January 2025}}《申报》1877年12月7日载：“今岁豫省之灾，亦不减于山右，……灾黎数百万，几有易子析骸之惨”&lt;/p&gt;
#  1900年，光绪二十六年：二十六年，两宫西狩，关中大饥，人相食，（唐）锡晋醵金四十万往赈，历二州八县，艰困不少阻。（《清史稿·卷四百五十二·列传二百三十九·洪汝奎等》㉕*）
#  1910年，宣统二年十二月：是月，江、淮饥，人相食。东三省疫。（《清史稿·卷二十五·本纪二十五·宣统皇帝本纪》㉕*）
#  1911年，宣统三年：钟麟同，字建堂，山东济宁州人。威海武备学堂毕业。……宣统三年九月初九日，七十三标兵变，夜半，自北校场入城。……以手枪自击而仆，变军碎其尸，剖心啖之。上闻，有“忠骸支解，惨不忍闻”之谕，谥忠壮。（《清史稿·卷四百九十六·列传二百八十三·忠义十》㉕*）
#  光熙，本名惠熙，字亮臣。少从盛昱游，励学。钟琦遘危疾，尝刲股和药以进。（《清史稿·卷四百六十九·列传二百五十六·恩铭等》㉕*）
#  礼堂，字和贵。事亲孝。父继宏，久疟，冬月畏火，礼堂潜以身温被。居丧如礼，笑不见齿。母遘危疾，刲股合药，私祷于神，减齿以延亲寿。（《清史稿·卷四百八十一·列传二百六十八·儒林二》㉕*）
#  宋大樽，字左彝，仁和人。弱岁，刲股愈母疾，让产其弟。（《清史稿·卷四百八十五·列传二百七十二·文苑二》㉕*）
#  潘德舆，字四农，山阳人。年五六岁，母病不食，亦不食。父咯血，刲臂肉和药进，父察其色动，泣曰：“固知儿有是也！”（《清史稿·卷四百八十六·列传二百七十三·文苑三》㉕*）
#  曾艾，字虎卿，湖南新化人。尝割左臂疗父疾。（《清史稿·卷四百八十九·列传二百七十六·忠义三》㉕*）
#  陈源兖，字岱云，湖南茶陵州人。道光十八年进士，改翰林，授编修，旋授江西吉安府。先是源兖妻易氏以源兖遘疾几殆，籥天原以身代，刲臂和药饮源兖，源兖以愈，易氏旋病卒。同乡公举孝妇，请旌于朝。（《清史稿·卷四百九十·列传二百七十七·忠义四》㉕*）
#  沈瀛，字士登，江苏吴县人。尝刲臂疗母疾。（《清史稿·卷四百九十六·列传二百八十三·忠义十》㉕*）
#  李盛山，福建罗源人。母病，割肝以救，伤重，卒。巡抚常赉疏请旌，下礼部，礼部议轻生愚孝，无旌表之例。雍正六年三月壬子，世宗谕曰：“……父母爱子，无所不至，若因己病而致其子割肝刲股以充饮馔、和汤药，纵其子无恙，父母未有不惊忧恻怛惨惕而不安者，况因此而伤生，岂父母所忍闻乎？父母有疾，固人子尽心竭力之时，傥能至诚纯孝，必且感天地、动鬼神，不必以惊世骇俗之为，著奇于日用伦常之外。……倘训谕之后，仍有不爱躯命，蹈于危亡者，朕亦不概加旌表，以成激烈轻生之习也。”盛山仍予旌表。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  吕斅孚，湖南永定人。父孟卿，贫，以客授自给。母病将殆，思肉食，斅孚方七岁，贷诸屠，屠不可，泣而归。闻母呻吟，益痛，内念股肉可啗母，取厨刀砺使利，割右股四寸许，授其女弟，方五岁，令就炉火炙以进。母疾良已，孟卿归，察斅孚足微跛，得其状，与母持以哭。斅孚曰：“毋然，儿固无所苦也。”……孟卿亦尝刲股愈父病，然斅孚割股时，初不知父有是事也。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  汪灏，江南休宁人。晨、日昂、日升，其弟也。父病咯血，灏年十六，割股和药进，良愈。后数年病足，晨割股炼为末，敷治亦愈。又数年复咯血，晨复割臂以疗。更数年，疾大作，灏复割臂，勿瘳。晨病，日昂泣曰：“吾兄割臂愈父，吾不能割以愈吾兄乎？”众尼之。懵且仆，匠治棺，日升持匠斧断指，血淋漓，调药以饮晨。有司表其门曰“一门四孝友”。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  觉罗色尔岱，满洲镶红旗人，德世库七世孙也。性笃孝。年十七，父病，医不效，乃割左臂为糜以进，病稍间，旋歾。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  康熙间，以割臂疗亲旌者，有翁杜、佟良，与色尔岱同时有克什布。翁杜，满洲镶白旗人；佟良，蒙古镶黄旗人：官防御。克什布，满洲镶红旗人，官三等侍卫。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  奚缉营，字圣辉，江苏宝山人。父士本，以孝旌。缉营幼读论语，至“父母之年，不可不知”，辄陨涕簌簌，师奇之，谓真孝子子也。母病，刲臂以疗。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  张三爱，江南歙县人。为人役。事母孝，母病，不能具药物。或谓之曰：“汝欲愈母病，盍刲肝？”三爱祷于丛祠，破腹，肝堕出，以右手劙肝，得指许，左手纳于腹，束以白麻。归以肝和羹饮母，母良愈，三爱创亦合。（《清史稿·卷四百九十七·列传二百八十四·孝义一》㉕*）
#  杨献恒，山东益都人。父加官，与济南杨开泰有隙，……开泰计必欲杀献恒，遣其子承恩至青州谋诸吏。献恒潜知之，持铁骨朵挟刃至所居。承恩方与吏耳语，伺其出，以铁骨朵击之，仆，急拔刀断其喉，又抉其睛啖之，诣县自陈，出所藏银为证。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  刘希向，江南山阳人。……父病，希向为割股，良愈。希向年六十，病噎，其子亦割股，刀钝，肉不决，剪之，乃下，然希向竟不瘳。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  县有嫠张陈氏，家贫，刲肉以奉姑，训予田十亩助其养。（《清史稿·卷四百九十八·列传二百八十五·孝义二》㉕*）
#  李孔昭，字光四，蓟州人。……崇祯十五年进士，……母病，刲股疗之。（《清史稿·卷五百一·列传二百八十八·遗逸二》㉕*）
#  萧学华妻贺，湖南安化人。贺父徙陕西，学华赘其家。年余，学华归省母，贺欲与俱，父不许，贺割股肉付夫以奉姑。姑適病，学华烹肉进，病良已。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  子日焜妻李，尝刲股愈母病，事祖姑及姑孝。姑病，割臂进，病目，舐以舌，良已。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  王钜妻施，钜，萧山人；施，富阳人。姑严，小不当意，辄呵斥，施屏息不敢声。姑病反胃甚，医以为不治，施刲股和药进，病良已，姑遇施如故。钜疾作，施视疾惫，病瘵卒，姑犹不善施。钜以刲股事告，视其尸，信，乃大恸曰：“吾负孝妇！”（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  陈文世妻刘，郧人。陈、刘皆农家，刘待年于陈。既婚，姑年七十二，病噎，刘割臂和药以进，疾少间；既而复作，不食已十日，垂尽矣。刘夜屏人，杀鸡誓于神，持小刀自劙其胸二寸许，出肝刲半，取布束创，以肝与鸡同瀹汤奉姑。姑久不言，忽曰：“汤香甚！”饮之竟，病良愈，刘亦旋平。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  林经妻陈，连江人，姑盲性卞，常臆妇藐己，陈断三指自明，姑为之悔。经病，刲股；经卒，以节终。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  林云铭妻蔡，云铭，闽人；……耿精忠反，下云铭狱，蔡忧之，呕血殷紫，女瑛佩剜臂肉入药，旋苏。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  崔龙见妻钱，名孟钿，字冠之，一字浣青。龙见，永济人；钱，武进人，侍郎维城女。九岁刲臂疗父疾。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  张茂信妻方，茂信，河津人；方仪徵人。方尝割股愈舅疾，舅与茂信皆卒，奉姑刘。（《清史稿·卷五百八·列传二百九十五·列女一》㉕*）
#  （袁）进忠病，疡生于胫，（养）女刲股以疗，家人皆不知，而长女虐愈甚。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  王前洛聘妻林，潜山人。前洛病，林父饣鬼药，林潜刲股入药。前洛卒，固请奔丧，引刀誓不嫁。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  徐文经聘妻姚，名淑金，侯官人。文经卒，淑金屡求死，乃归于徐。贫，舅殁，姑疾作，刲股以疗。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  乔涌涛聘妻方，桐城人。涌涛卒，涌涛母丁亦病，方请于父母，归于乔。以姑病寒疾，亦薄其衣当风雪。刲股以进姑，病良已。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  袁绩懋妻左，绩懋见《忠义传》。左名锡璇，字芙江，阳湖人。事亲孝，父病，刲臂和药进。工诗善画，书法尤精，著有卷葹阁诗集。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  何其仁聘妻李，路南人。嘉庆十一年，年十六，未行。其仁及其父皆病笃，李割股畀叔母使送婿家。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  林国奎妻郑，闽人。国奎卒，有子二。郑将殉，姑诫以存孤，乃已。一子殇，遂自沉于江，渔者拯以还。姑疾，刲肝杂糜进，疾良已。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  吉山妻瓜尔佳氏，名惠兴，满洲人，杭州驻防。早寡，事姑谨，尝刲肱疗姑疾。（《清史稿·卷五百九·列传二百九十六·列女二》㉕*）
#  王如义妻向，涪州人。幼能为诗文。如义，农家子，向恒劝之读。道光十六年，如义暴卒，姑喻之嫁，矢以死。舅病，为刲股。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  许会妻张，颍州人。姑姣而虐，恶张端谨不类，日诟且挞，张事姑益恭。姑病，刲股以疗，姑虐如故。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  安于磐妻朱、后妻田，于磐，贵州蛮夷司长官。初娶朱，事姑孝，姑病，刲股，卒。复娶田，于磐病，刲股。于磐卒，抚诸子成立。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  田养民妻杨，养民，朗溪司长官；杨，邑梅司人也。年十二，母病，刲股。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  伊嵩阿，拜都氏，满洲镶黄旗人；妻希光，钮祜禄氏，正白旗人，总督爱必达女也。伊嵩阿为大学士永贵从子，早卒。方病时，希光割股进，终不起，许以死。爱必达、永贵共喻之，誓毕婚嫁乃殉。为伊嵩阿弟娶，嫁女妹及二女，次女行之明日，自缢死。张遗诗于壁，略谓：“十载要盟，此日当报命。”乾隆四十六年三月事也。永贵疏闻，高宗为赋诗，旌其节。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）
#  朱承宇妻曹，承宇，无锡人；曹，武进人：皆农家也。生二子、一女，而承宇死。承宇弟迫之嫁，曹以死拒。……哭于承宇墓，还，遂缢。……及敛，左臂创未合，盖承宇病时尝割臂也。父为讼于县，罪迫嫁者。（《清史稿·卷五百十一·列传二百九十八·列女四》㉕*）

==中华民国==
1936年“3月1日万源曹家沟某家七人，饿毙四人；余三人气息奄奄，竟为逃荒饥民杀死，分割炙食无余。”{{cfn|许汉三|y=1985}}
1936年3月19日四川省报载：“北川县人肉每斤五百文。片口镇饥民张彭氏、何张氏、陈顺氏因饥饿难忍，挖掘死尸围食，被捕。”{{cfn|许汉三|y=1985}}

1936年四川《民间意识》杂志汇载四川各地吃人的消息：“松潘半边街居民陈氏，自杀其八岁的亲生女而食，食尽仍病饿而死。沿途数百里内，人血、白骨与饿死者，填满沟壑。”{{cfn|许汉三|y=1985}}

民國30年（1941）-民國32年（1943）河南省大旱，人相食。1942年河南省赈济会推选[[:w:杨一峰|杨一峰]]、[[:w:刘庄甫|刘庄甫]]、[[:w:任兆鲁|任兆鲁]]三人等赴[[:w:重庆|重庆]]，请国民党中央免除徵賦，蒋介石拒不接见。大公报主笔[[:w:王芸生|王芸生]]在1942年的一篇《看重庆，念中原》的社论中写道：“饿死的暴骨失肉，逃亡的扶老携幼，妻离子散，挤人丛，挨棍打，未必能够得到赈济委员会的登记证。吃杂草的毒发而死，吃干树皮的忍不住刺喉绞肠之苦。把妻女驮运到遥远的人肉市场，未必能够换到几斗粮食。”[[:w:冯小刚|冯小刚]]於2012年拍摄的电影《一九四二》讲的正是这段时期发生的故事。

1948年6月[[:w:國共內戰|國共內戰]]期間，[[:w:中共|中共]]将领[[:w:林彪|林彪]]進行[[:w:長春圍城|長春圍城]]，禁止糧食進城，國軍于是收集城內的糧食，造成很多人餓死街頭。10月21日，城內守軍[[:w:鄭洞國|鄭洞國]]投降。活過來的人說，「就喝死人腦瓜殼裡的水，都是蛆。就這麼熬著，盼著，盼開卡子放人。就那麼幾步遠，就那麼瞅著，等人家一句話放生。卡子上天天宣傳，說誰有槍就放誰出去。真有有槍的，真放，交上去就放人。每天都有，都是有錢人，在城裡買了準備好的，都是手槍。咱不知道。就是知道，哪有錢買呀！」參加圍城的中共官兵說：「在外邊就聽說城裡餓死多少人，還不覺怎麼的。從死人堆裡爬出多少回了，見多了，心腸硬了，不在乎了。可進城一看那樣子就震驚了，不少人就流淚了。」&lt;ref&gt;张正隆：《雪白血红》&lt;/ref&gt;

==中華人民共和國==

=== 三年大跃进时期 ===
1959年－1961年「[[:w:大跃进|大躍進]]」期間，中國大陸發生“[[:w:三年困难时期|三年大饑荒]]”，据各方估计共造成1500万-5500万[[:w:非正常死亡|非正常死亡]]&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|title=The Institutional Causes of China's Great Famine, 1959–1961|author=|url=https://www0.gsb.columbia.edu/faculty/pyared/papers/famines.pdf|first1=XIN|last2=QIAN|first2=NANCY|date=2015-01|journal=Review of Economic Studies|issue=4|doi=10.1093/restud/rdv016|others=|year=|volume=82|page=|pages=1568–1611|pmid=|last3=YARED|first3=PIERRE|archive-date=2019-09-06|url-status=|via=|last1=MENG|archive-url=https://web.archive.org/web/20190906163322/https://www0.gsb.columbia.edu/faculty/pyared/papers/famines.pdf|dead-url=no}}&lt;/ref&gt;&lt;ref name=&quot;:29&quot;&gt;{{Cite web|title=西方学术界的大跃进饥荒研究|url=http://ww2.usc.cuhk.edu.hk/PaperCollection/webmanager/wkfiles/2012/201503_38_paper.pdf|author=陈意新|date=2015-01|format=|work=[[:w:香港中文大学|香港中文大学]]|publisher=《江苏大学学报》|language=zh|archive-url=https://web.archive.org/web/20210517052743/http://ww2.usc.cuhk.edu.hk/PaperCollection/webmanager/wkfiles/2012/201503_38_paper.pdf|archive-date=2021-05-17|dead-url=no}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|title=SITES OF HORROR: MAO'S GREAT FAMINE [with Response]|author=Felix Wemheuer|url=http://www.jstor.org/stable/41262812|date=2011|journal=The China Journal|issue=66|doi=|others=|year=|editor-last=Dikötter|editor-first=Frank|volume=|page=|pages=155–164|issn=1324-9347|pmid=|archive-url=https://web.archive.org/web/20200727141524/https://www.jstor.org/stable/41262812|archive-date=2020-07-27|dead-url=no}}&lt;/ref&gt;。餓殍遍野，到處都有餓死倒斃在路邊的人，有些地方甚至出現吃人肉的現象。[[:w:楊繼繩|杨继绳]]所著的《[[:w:墓碑 (书籍)|墓碑]]》一書援引梁志遠的《關於「特種案件」的匯報——安徽亳縣人吃人見聞錄》記載指人吃人並不是個別現象：“其面積之廣，數量之多，時間之長，實屬世人罕見”{{cfn|楊繼繩|y=2008|p=274}}。

1960年春，吃人肉情況不斷發生，人肉的交易市場也隨之出現在城郊、集鎮、農民擺攤等{{cfn|楊繼繩|y=2008|p=278}}。三年大饑荒的[[:w:口述歷史|口述歷史]]《[[:w:尋找大饑荒倖存者|尋找大饑荒倖存者]]》记载了四十九起人吃人事件&lt;ref name=&quot;rfa&quot;&gt;{{Cite news|url=https://www.rfa.org/mandarin/pinglun/wenyitiandi-cite/yujie-01082014125845.html|title=为当代中国修筑一面“哭墙”--依娃《寻找大饥荒幸存者》|publisher=[[:w:自由亚洲电台|自由亚洲电台]]|date=2014-01-08|archive-url=https://web.archive.org/web/20210722001314/https://www.rfa.org/mandarin/pinglun/wenyitiandi-cite/yujie-01082014125845.html|archive-date=2020-07-22|dead-url=no|language=zh|author=余杰|authorlink=余杰}}&lt;/ref&gt;。人吃人事件在[[:w:四川|四川]]、[[:w:甘肅|甘肅]]、[[:w:青海|青海]]、[[:w:西藏|西藏]]、[[:w:陝西|陝西]]、[[:w:寧夏|寧夏]]、[[:w:河北|河北]]、[[:w:遼寧|遼寧]]皆有耳聞，幾乎遍及全國{{cfn|貝克|y=2005}}。據作家[[:w:沙青|沙青]]的[[:w:报告文学|報告文學]]記載：「有一戶農家，吃得只剩了父親和一男一女兩個孩子。一天，父親將女兒趕出門去，等女孩回家時，弟弟不見了，鍋裡浮著一層白花花油乎乎的東西，灶邊扔著一具骨頭。幾天之後，父親又往鍋裡添水，然後招呼女兒過去。女孩嚇得躲在門外大哭，哀求道：『爸爸，別吃我，我給你摟草、燒火，吃了我沒人給你做活。』」&lt;ref&gt;{{Cite web|title=依稀大地湾——大饥荒年代|url=https://boxun.com/news/gb/z_special/2004/12/200412281348.shtml?__cf_chl_jschl_tk__=pmd_cf65954eb189551663c797db8d490efde1f84d97-1626912600-0-gqNtZGzNAg2jcnBszQti|author=沙青|date=2004-12-28|publisher=[[:w:博讯|博讯]]|language=zh|archive-url=https://web.archive.org/web/20080822033646/http://www.peacehall.com/news/gb/z_special/2004/12/200412281348.shtml|archive-date=2008-08-22|dead-url=no}}&lt;/ref&gt;

* '''四川'''：《[[:w:中國大饑荒，1958-1962|中國大饑荒，1958-1962]]》引用的中國官方檔案中有吃人記載，如在[[:w:四川省|四川省]][[:w:石柱土家族自治縣|石柱土家族自治縣]]的桥头区，老妇人罗文秀是第一个开始吃人肉的人。在家人一家七口全部死去后，罗文秀把三岁女童马发慧的尸体挖出来。她把小女孩儿的肉割下来，用辣椒调味，然后蒸熟吃掉&lt;ref name=&quot;紐約時報&quot;&gt;{{cite news|url=http://cn.nytimes.com/china/20120917/c17famine/|title=記錄大饑荒人相食的慘劇|publisher=《[[:w:紐約時報|紐約時報]]》|date=2012年9月17日|archive-date=2013年10月23日|archive-url=https://web.archive.org/web/20131023013637/http://cn.nytimes.com/china/20120917/c17famine/|dead-url=no|author=DIDI KIRSTEN TATLOW|language=zh}}&lt;/ref&gt;。另一份1961年1月27日的文件，讲述了一个四川母亲用毛巾勒死了自己五岁大的儿子，“吃了四顿”。调查者王德明写道，“这样令人震惊的可怕事件远非只有这一起。”&lt;ref name=&quot;紐約時報&quot; /&gt;

* '''河南'''：1959年10月至1960年4月，[[:w:信阳事件|信陽事件]]，[[:w:商丘|商丘]]、[[:w:開封|開封]]餓得人身浮腫，吃樹皮，餓死100萬（到數百萬）人口，時諺：“人吃人，狗吃狗，老鼠餓得啃磚頭。”“信陽五里店村一個14、15歲的小女孩，将4、5歲的弟弟殺死煮了吃了，因爲父母都餓死了，只剩下這兩個孩子，女孩餓得不行，就吃弟弟。”{{cfn|楊繼繩|y=2008}} 河南省[[:w:固始县|固始縣]]官方記載有二百例人吃人事件，縣委以“破壞屍體”為名，逮捕群眾{{cfn|貝克|y=2005|p=180|url=https://books.google.com/books?id=hjpdAAAAIAAJ&amp;q=固始縣+二百}}。鹿邑、夏邑、虞城、永城等县共发现吃死人肉的情况20多起。据中央工作组魏震报告，鹿邑县从1959年10月到1960年11月，发现人吃人的事件6起。马庄公社马庄大队庞王庄18岁女子王玉娥于1960年4月19日将堂弟弟5岁的王怀郎溺死煮食，怀郎14岁的亲姐姐小朋也因饥饿吃了弟弟的肉。&lt;ref&gt;{{cite news |title=[杨继绳]《墓碑》――中国六十年代大饥荒纪实. |url=http://|publisher=第54頁 |accessdate=2022-03-23}}&lt;/ref&gt;

* '''甘肃'''：[[:w:通渭县|通渭縣]]，1958年全縣糧食實產8300多萬斤，虛報1.8億斤。人口大量死亡；有人回憶“1959年11月到臘月，死的人多。老百姓一想那事就要流淚。餓死老人家的，餓死婆娘的，日子過得糊裡糊塗。把人煮了吃，肉割來煮了吃……人甚麼也不想，甚麼也不怕，就想吃，想活。把娃娃、自己的娃娃吃下的，也有；把外面逃到村上的人殺了吃的，也有。吃下自己娃娃的，浮腫，中毒，不像人樣子。有的病死了，也有救下的。吃了娃娃心裡慘的，吃過就後悔了，自己恨自己。在村子里住不下去，沒人理他，嫌他臟。”（《50年代末大飢荒驚人記實》）
* '''青海'''：人吃人事件110多起，漢東公社楊家灘生產隊的婦女竟吃了9個小孩&lt;ref&gt;武文軍:《餓魂祭:中國六十年代饑荒考》，蘭州學刊2005年專輯，蘭州社會科學院主編，p110-110&lt;/ref&gt;。
* '''湖南'''：据余习广《吃人饿鬼：[[:w:刘家远惨杀亲子食子案|刘家远惨杀亲子食子案]]》記載，[[:w:湖南|湖南]][[:w:澧县|澧县]]如东公社男子刘家远，將自己儿子殺害後烹煮食用。刘家远也因食子而被處決&lt;ref&gt;{{cite news|title=毛泽东时代惨剧：三年大饥荒饥民十大奇吃|url=https://www.dwnews.com/%E4%B8%AD%E5%9B%BD/59674203/%E6%AF%9B%E6%B3%BD%E4%B8%9C%E6%97%B6%E4%BB%A3%E6%83%A8%E5%89%A7%E4%B8%89%E5%B9%B4%E5%A4%A7%E9%A5%A5%E8%8D%92%E9%A5%A5%E6%B0%91%E5%8D%81%E5%A4%A7%E5%A5%87%E5%90%83|publisher=[[:w:共识网|共识网]]|archive-date=2020-11-05|archive-url=https://web.archive.org/web/20201105165243/https://www.dwnews.com/%E4%B8%AD%E5%9B%BD/59674203/%E6%AF%9B%E6%B3%BD%E4%B8%9C%E6%97%B6%E4%BB%A3%E6%83%A8%E5%89%A7%E4%B8%89%E5%B9%B4%E5%A4%A7%E9%A5%A5%E8%8D%92%E9%A5%A5%E6%B0%91%E5%8D%81%E5%A4%A7%E5%A5%87%E5%90%83|dead-url=no|author=惠风（原作者：彭劲秀）|date=2014-03-11|language=zh|agency=[[:w:多維新聞|多維新聞]]}}&lt;/ref&gt;。
* '''安徽'''：作家[[:w:王立新 (1949年)|王立新]]1980年代曾赴[[:w:凤阳县|凤阳]]采访过，他在报告文学中写道：“梨园乡小岗生产队严俊冒告诉我：1960年，我们村附近有个死人塘，浮埋着许多饿死的人。为什么浮埋？饿得没力气呀，扔几锹土了事。说起来，对不起祖先，也对不起冤魂。人饿极了，什么事都干得出来。我的一位亲戚见人到死人塘割死人的腿肚子吃，她也去了。开始有点怕，后来惯了，顶黑去顶黑回。我问她：‘怎么能……？’她叹息道：‘饿极了。’”&lt;ref&gt;[[:w:李锐 (1917年)|李锐]]《大跃进亲历记》（南方出版社1999年版）&lt;/ref&gt;

=== 文化大革命时期 ===
{{main|:w:广西文革屠杀}}

[[:w:文化大革命|文化大革命]]時期（1966-1976年），[[:w:广西壮族自治区|广西壮族自治区]]除[[:w:广西文革屠杀|私刑、屠杀事件众多]]外，亦傳出多起食人事件&lt;ref name=&quot;:13&quot;&gt;{{Cite web|title=不反思“文革”的社会，就是个食人部落|url=http://history.people.com.cn/n/2013/0305/c200623-20680503.html|author=[[:w:张鸣 (学者)|张鸣]]|date=2013-03-05|format=|work=|publisher=《[[:w:中国青年报|中国青年报]]》|agency=[[:w:人民网|人民网]]|language=zh|archiveurl=https://web.archive.org/web/20200625141907/http://history.people.com.cn/n/2013/0305/c200623-20680503.html|archivedate=2020-06-25|dead-url=yes}}&lt;/ref&gt;&lt;ref name=&quot;:0&quot;&gt;{{Cite web|title=我参与处理广西文革遗留问题|url=http://www.yhcqw.com/34/8938.html|accessdate=2019-11-29|author=晏乐斌|date=|format=|work=|publisher=《[[:w:炎黄春秋|炎黄春秋]]》|language=zh|archive-url=https://web.archive.org/web/20191207031844/http://www.yhcqw.com/34/8938.html|archive-date=2019-12-07|dead-url=yes}}&lt;/ref&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite web|title=广西文革中的吃人狂潮|url=http://www.cuhk.edu.hk/ics/21c/media/articles/c155-201605003.pdf|accessdate=|author=[[:w:宋永毅|宋永毅]]|date=|format=|publisher=[[:w:香港中文大学|香港中文大学]]|language=zh|archive-url=https://web.archive.org/web/20180127184237/http://www.cuhk.edu.hk/ics/21c/media/articles/c155-201605003.pdf|archive-date=2018-01-27|dead-url=no}}&lt;/ref&gt;。作家[[:w:鄭義 (作家)|鄭義]]曾在文革後赴廣西調查，于1993年出版《[[:w:红色纪念碑|红色纪念碑]]》一书，據他的統計廣西全省至少有一千人被食。紀錄片「文革廣西[[:w:武宣县|武宣縣]]紅衛兵吃人肉事件」評論称：“這些食人事件並不是因為飢荒，而是因為政治運動製造出來的仇恨心態&lt;ref&gt;{{Cite web |url=https://www.youtube.com/watch?v=vR2JhwcEM1A |title=文革廣西武宣縣紅衛兵吃人肉事件 |accessdate=2015-07-25 |archive-date=2016-03-16 |archive-url=https://web.archive.org/web/20160316105309/https://www.youtube.com/watch?v=vR2JhwcEM1A |dead-url=no }}&lt;/ref&gt;”。

其中人食人最厲害的地方之一是廣西[[:w:武宣县|武宣縣]]，官方调查发现至少38人被吃&lt;ref name=&quot;:0&quot; /&gt;，民间研究调查则发现有70余人&lt;ref name=&quot;:4&quot; /&gt;甚至上百人被吃&lt;ref name=&quot;:12&quot;&gt;{{Cite web|title=Chronology of Mass Killings during the Chinese Cultural Revolution (1966-1976)|url=https://www.sciencespo.fr/mass-violence-war-massacre-resistance/en/document/chronology-mass-killings-during-chinese-cultural-revolution-1966-1976|accessdate=|author=[[:w:宋永毅|宋永毅]]|date=2011-08-25|format=|publisher=[[:w:巴黎政治学院|巴黎政治学院]]（Sciences Po）|language=en|archive-url=https://web.archive.org/web/20190425062821/https://www.sciencespo.fr/mass-violence-war-massacre-resistance/en/document/chronology-mass-killings-during-chinese-cultural-revolution-1966-1976|archive-date=2019-04-25|dead-url=no}}&lt;/ref&gt;。武宣县“一女民兵因参与杀人坚定勇敢，且专吃男人生殖器而臭名远播，并因此入党做官，官至武宣县革委副主任。处遗时期中共中央书记处一天一个电话催问处理结果，并严厉责问：‘像这样的人，为什么还不赶快开除党籍？’但该副主任拒不承认专吃生殖器，只承认一起吃过人。最后的处理是开除党籍，撤销领导职务。现已调离武宣。”{{cfn|鄭義|y=1993|p=74-75|url=https://books.google.com/books?id=IJBxAAAAIAAJ&amp;q=武宣縣+副主任}}

== 参考文献 ==
=== 引用 ===
{{Reflist|30em}}

=== 来源 ===
{{refbegin}}
* 王永寬：《中國古代酷刑》
* [[:w:黃文雄 (作家)|黃文雄]]：《中國食人史》
* 黃粹涵：《中國食人史料鈔》
* {{cite book
|author=许汉三
|title=《黃炎培年谱》
|url=https://books.google.com/books?id=z2djAAAAIAAJ
|year=1985年
|publisher=文史资料出版社
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-26
|archive-url=https://web.archive.org/web/20210426094608/https://books.google.com/books?id=z2djAAAAIAAJ
}}
* {{cite book
|author=鄭義
|title=《紅色紀念碑》
|url=https://books.google.com/books?id=IJBxAAAAIAAJ
|year=1993年
|publisher=華視文化
|isbn=978-957-572-048-3
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-26
|archive-url=https://web.archive.org/web/20210426200250/https://books.google.com/books?id=IJBxAAAAIAAJ
}}
* {{cite book
|author=楊繼繩
|author-link=楊繼繩
|title=《墓碑——中國六十年代大饑荒紀實 上篇》
|url=https://books.google.com/books?id=GnglAQAAMAAJ
|year=2008年
|publisher=天地圖書
|isbn=978-988-211-909-3
|ref=harv
|access-date=2021-04-19
|archive-date=2021-04-19
|archive-url=https://web.archive.org/web/20210419003552/https://books.google.com/books?id=GnglAQAAMAAJ
}}
* {{cite book 
| author=賈斯柏‧貝克 
| translator=姜和平  
| title=《餓鬼：毛時代大饑荒揭秘》
| publisher=明鏡出版社 
| date=2005年10月 
| url=http://books.google.com/books?id=hjpdAAAAIAAJ 
| isbn=978-1-932138-30-6
| ref = {{SfnRef|貝克|2005}}}}
* [[:w:有線電視|有線電視]]財經資訊台《神州穿梭》 「文革廣西武宣縣紅衛兵吃人肉事件」
{{refend}}

== 外部链接 ==
*[[:w:钱理群|钱理群]]：《[http://www.aisixiang.com/data/3951-2.html 钱理群：说“食人”——周氏兄弟改造国民性思想之一]》{{Wayback|url=http://web.archive.org/web/20150605170543/http://www.aisixiang.com/data/3951-2.html |date=20150605170543 }}

[[Category:History of China]]</text>
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